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-11400 | Area inside Astroid | The area inside an astroid $H$ constructed within a circle of radius $a$ is given by:
:$\AA = \dfrac {3 \pi a^2} 8$ | Let $H$ be embedded in a cartesian plane with its center at the origin and its cusps positioned on the axes.
:400px
By symmetry, it is sufficient to evaluate the area shaded yellow and to multiply it by $4$.
By Equation of Astroid:
:$\begin{cases}
x & = a \cos^3 \theta \\
y & = a \sin^3 \theta
\end{cases}$
Thus:
{{begi... | The [[Definition:Area|area]] inside an [[Definition:Astroid|astroid]] $H$ constructed within a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ is given by:
:$\AA = \dfrac {3 \pi a^2} 8$ | Let $H$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its center at the [[Definition:Origin|origin]] and its [[Definition:Cusp of Hypocycloid|cusps]] positioned on the [[Definition:Coordinate Axis|axes]].
:[[File:AstroidArea.png|400px]]
By [[Definition:Symmetry|symmetry]], it is sufficient to... | Area inside Astroid | https://proofwiki.org/wiki/Area_inside_Astroid | https://proofwiki.org/wiki/Area_inside_Astroid | [
"Astroids"
] | [
"Definition:Area",
"Definition:Astroid",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Hypocycloid/Cusp",
"Definition:Axis/Coordinate Axes",
"File:AstroidArea.png",
"Definition:Symmetry",
"Definition:Area",
"Equation of Astroid",
"Definition:Differentiation",
"Definition:Integration/Integrand",
"Doubl... |
proofwiki-11401 | Length of Arc of Astroid | The total length of the arcs of an astroid constructed within a deferent of radius $a$ is given by:
:$\LL = 6 a$ | Let $H$ be embedded in a cartesian plane with its center at the origin and its cusps positioned on the axes.
:400px
We have that $\LL$ is $4$ times the length of one arc of the astroid.
From Arc Length for Parametric Equations:
:$\ds \LL = 4 \int_{\theta \mathop = 0}^{\theta \mathop = \pi/2} \sqrt {\paren {\frac {\d x}... | The total [[Definition:Arc Length|length of the arcs]] of an [[Definition:Astroid|astroid]] constructed within a [[Definition:Deferent of Hypocycloid|deferent]] of [[Definition:Radius of Circle|radius]] $a$ is given by:
:$\LL = 6 a$ | Let $H$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its center at the [[Definition:Origin|origin]] and its [[Definition:Cusp of Hypocycloid|cusps]] positioned on the [[Definition:Coordinate Axis|axes]].
:[[File:Astroid.png|400px]]
We have that $\LL$ is $4$ times the [[Definition:Arc Length|... | Length of Arc of Astroid | https://proofwiki.org/wiki/Length_of_Arc_of_Astroid | https://proofwiki.org/wiki/Length_of_Arc_of_Astroid | [
"Astroids"
] | [
"Definition:Arc Length",
"Definition:Astroid",
"Definition:Hypocycloid/Generator/Deferent",
"Definition:Circle/Radius"
] | [
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Hypocycloid/Cusp",
"Definition:Axis/Coordinate Axes",
"File:Astroid.png",
"Definition:Arc Length",
"Definition:Hypocycloid/Arc",
"Definition:Astroid",
"Arc Length for Parametric Equations",
"Equation of Astroid",
"S... |
proofwiki-11402 | Area of Surface of Revolution from Astroid | Let $H$ be the astroid constructed within a circle of radius $a$.
The surface of revolution formed by rotating $H$ around the $x$-axis:
:400px
evaluates to:
:$\SS = \dfrac {12 \pi a^2} 5$ | By symmetry, it is sufficient to calculate the surface of revolution of $H$ for $0 \le x \le a$.
From Area of Surface of Revolution, this surface of revolution is given by:
:$\ds \SS = 2 \int_0^{\pi / 2} 2 \pi y \, \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
From Equa... | Let $H$ be the [[Definition:Astroid|astroid]] constructed within a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$.
The [[Definition:Surface of Revolution|surface of revolution]] formed by rotating $H$ around the [[Definition:X-Axis|$x$-axis]]:
:[[File:AstroidSurfaceOfRevolution.png|400px]... | By [[Definition:Symmetry|symmetry]], it is sufficient to calculate the [[Definition:Surface of Revolution|surface of revolution]] of $H$ for $0 \le x \le a$.
From [[Area of Surface of Revolution]], this [[Definition:Surface of Revolution|surface of revolution]] is given by:
:$\ds \SS = 2 \int_0^{\pi / 2} 2 \pi y \, \s... | Area of Surface of Revolution from Astroid | https://proofwiki.org/wiki/Area_of_Surface_of_Revolution_from_Astroid | https://proofwiki.org/wiki/Area_of_Surface_of_Revolution_from_Astroid | [
"Astroids",
"Examples of Surfaces of Revolution"
] | [
"Definition:Astroid",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Surface of Revolution",
"Definition:Axis/X-Axis",
"File:AstroidSurfaceOfRevolution.png"
] | [
"Definition:Symmetry",
"Definition:Surface of Revolution",
"Area of Surface of Revolution",
"Definition:Surface of Revolution",
"Equation of Astroid",
"Sum of Squares of Sine and Cosine",
"Primitive of Power of Sine of a x by Cosine of a x"
] |
proofwiki-11403 | Tusi Couple is Diameter of Deferent | A Tusi couple is a degenerate case of the hypocycloid whose form is a straight line that forms a diameter of the deferent. | Let $C_1$ be a circle of radius $b$ rolling without slipping around the inside of a circle $C_2$ of radius $a$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference of $C_1$.
Let $C_1$ be initially positioned so that $P$ is its point of tangency t... | A [[Definition:Tusi Couple|Tusi couple]] is a [[Definition:Degenerate Case|degenerate case]] of the [[Definition:Hypocycloid|hypocycloid]] whose form is a [[Definition:Straight Line|straight line]] that forms a [[Definition:Diameter of Circle|diameter]] of the [[Definition:Deferent of Hypocycloid|deferent]]. | Let $C_1$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $b$ rolling without slipping around the inside of a [[Definition:Circle|circle]] $C_2$ of [[Definition:Radius of Circle|radius]] $a$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:Cen... | Tusi Couple is Diameter of Deferent | https://proofwiki.org/wiki/Tusi_Couple_is_Diameter_of_Deferent | https://proofwiki.org/wiki/Tusi_Couple_is_Diameter_of_Deferent | [
"Tusi Couples"
] | [
"Definition:Tusi Couple",
"Definition:Degenerate Case",
"Definition:Hypocycloid",
"Definition:Line/Straight Line",
"Definition:Circle/Diameter",
"Definition:Hypocycloid/Generator/Deferent"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Tangent Circles",
"Definition:Poin... |
proofwiki-11404 | Meet Preserves Directed Suprema | Let $\mathscr S = \struct {S, \preceq}$ be an up-complete meet semilattice such that
:$\forall x \in S$, a directed subset $D$ of $S$: $x \preceq \sup D \implies x \preceq \sup \set {x \wedge y: y \in D}$ | === Lemma 2 ===
{{:Meet Preserves Directed Suprema/Lemma 2}}{{qed|lemma}}
Let $X$ be a directed subset of $S \times S$ such that
:$X$ admits a supremum.
By Up-Complete Product:
:the simple order product of $\mathscr S$ and $\mathscr S$ is up-complete.
By Up-Complete Product/Lemma 2:
:$X_1 := \map {\pr_1^\to} X$ is dire... | Let $\mathscr S = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]] such that
:$\forall x \in S$, a [[Definition:Directed Subset|directed subset]] $D$ of $S$: $x \preceq \sup D \implies x \preceq \sup \set {x \wedge y: y \in D}$ | === [[Meet Preserves Directed Suprema/Lemma 2|Lemma 2]] ===
{{:Meet Preserves Directed Suprema/Lemma 2}}{{qed|lemma}}
Let $X$ be a [[Definition:Directed Subset|directed subset]] of $S \times S$ such that
:$X$ admits a [[Definition:Supremum of Set|supremum]].
By [[Up-Complete Product]]:
:the [[Definition:Simple Order ... | Meet Preserves Directed Suprema | https://proofwiki.org/wiki/Meet_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Meet_Preserves_Directed_Suprema | [
"Meet-Continuous Lattices",
"Up-Complete Semilattices"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Directed Subset"
] | [
"Meet Preserves Directed Suprema/Lemma 2",
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Up-Complete Product",
"Definition:Simple Order Product",
"Definition:Up-Complete",
"Up-Complete Product/Lemma 2",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Projecti... |
proofwiki-11405 | Equation of Astroid/Parametric Form | The point $P = \tuple {x, y}$ is described by the parametric equation:
:<nowiki>$\begin{cases}
x & = a \cos^3 \theta \\
y & = a \sin^3 \theta
\end{cases}$</nowiki>
where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$. | By definition, an astroid is a hypocycloid with $4$ cusps.
:400px
By Equation of Hypocycloid, the equation of $H$ is given by:
:<nowiki>$\begin{cases}
x & = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta} \\
y & = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}
\e... | The [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the [[Definition:Parametric Equation|parametric equation]]:
:<nowiki>$\begin{cases}
x & = a \cos^3 \theta \\
y & = a \sin^3 \theta
\end{cases}$</nowiki>
where $\theta$ is the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the ... | By definition, an [[Definition:Astroid|astroid]] is a [[Definition:Hypocycloid|hypocycloid]] with $4$ [[Definition:Cusp of Hypocycloid|cusps]].
:[[File:Astroid.png|400px]]
By [[Equation of Hypocycloid]], the equation of $H$ is given by:
:<nowiki>$\begin{cases}
x & = \paren {a - b} \cos \theta + b \map \cos {\paren {... | Equation of Astroid/Parametric Form | https://proofwiki.org/wiki/Equation_of_Astroid/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Astroid/Parametric_Form | [
"Astroids"
] | [
"Definition:Point",
"Definition:Parametric Equation",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center"
] | [
"Definition:Astroid",
"Definition:Hypocycloid",
"Definition:Hypocycloid/Cusp",
"File:Astroid.png",
"Equation of Hypocycloid",
"Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii",
"Definition:Hypocycloid/Generator/Epicycle",
"Definition:Circle/Radius",
"Definition:Circle/Radius",
"... |
proofwiki-11406 | Equation of Astroid/Cartesian Form | The point $P = \tuple {x, y}$ is described by the equation:
:$x^{2/3} + y^{2/3} = a^{2/3}$ | By definition, an astroid is a hypocycloid with $4$ cusps.
:400px
From the parametric form of the equation of an astroid, $H$ can be expressed as:
:<nowiki>$\begin{cases}
x & = 4 b \cos^3 \theta = a \cos^3 \theta \\
y & = 4 b \sin^3 \theta = a \sin^3 \theta
\end{cases}$</nowiki>
Squaring, taking cube roots and adding:
... | The [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the equation:
:$x^{2/3} + y^{2/3} = a^{2/3}$ | By definition, an [[Definition:Astroid|astroid]] is a [[Definition:Hypocycloid|hypocycloid]] with $4$ [[Definition:Cusp of Hypocycloid|cusps]].
:[[File:Astroid.png|400px]]
From the [[Equation of Astroid/Parametric Form|parametric form of the equation of an astroid]], $H$ can be expressed as:
:<nowiki>$\begin{cases... | Equation of Astroid/Cartesian Form | https://proofwiki.org/wiki/Equation_of_Astroid/Cartesian_Form | https://proofwiki.org/wiki/Equation_of_Astroid/Cartesian_Form | [
"Astroids"
] | [
"Definition:Point"
] | [
"Definition:Astroid",
"Definition:Hypocycloid",
"Definition:Hypocycloid/Cusp",
"File:Astroid.png",
"Equation of Astroid/Parametric Form",
"Definition:Square/Function",
"Definition:Cube Root",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-11407 | Preceding iff Meet equals Less Operand | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $x, y \in S$.
Then
:$x \preceq y$ {{iff}} $x \wedge y = x$ | === Sufficient Condition ===
Let
:$x \preceq y$
By definition of meet:
:$x \wedge y = \inf \set {x, y}$
By definitions of lower bound and reflexivity:
:$x$ is lower bound for $\set {x, y}$
and
:$\forall z \in S: z$ is lower bound for $\set {x, y} \implies z \preceq x$
Thus by definition of infimum:
:$x = \inf \set {x, ... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $x, y \in S$.
Then
:$x \preceq y$ {{iff}} $x \wedge y = x$ | === Sufficient Condition ===
Let
:$x \preceq y$
By definition of [[Definition:Meet (Order Theory)|meet]]:
:$x \wedge y = \inf \set {x, y}$
By definitions of [[Definition:Lower Bound of Set|lower bound]] and [[Definition:Reflexivity|reflexivity]]:
:$x$ is [[Definition:Lower Bound of Set|lower bound]] for $\set {x, y}... | Preceding iff Meet equals Less Operand | https://proofwiki.org/wiki/Preceding_iff_Meet_equals_Less_Operand | https://proofwiki.org/wiki/Preceding_iff_Meet_equals_Less_Operand | [
"Join and Meet",
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice"
] | [
"Definition:Meet (Order Theory)",
"Definition:Lower Bound of Set",
"Definition:Reflexivity",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set"
] |
proofwiki-11408 | Simple Harmonic Motion of Point on Tusi Couple | Let $C_1$ and $C_2$ be the epicycle and deferent respectively of a Tusi couple $H$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let the center of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$.
Let $P$ be the point on the circumference of $C_1$ wh... | {{ProofWanted|Straightforward but tedious. I'll get back to it in due course.}} | Let $C_1$ and $C_2$ be the [[Definition:Epicycle of Hypocycloid|epicycle]] and [[Definition:Deferent of Hypocycloid|deferent]] respectively of a [[Definition:Tusi Couple|Tusi couple]] $H$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:Center of Circle|center]] $O$ loca... | {{ProofWanted|Straightforward but tedious. I'll get back to it in due course.}} | Simple Harmonic Motion of Point on Tusi Couple | https://proofwiki.org/wiki/Simple_Harmonic_Motion_of_Point_on_Tusi_Couple | https://proofwiki.org/wiki/Simple_Harmonic_Motion_of_Point_on_Tusi_Couple | [
"Tusi Couples",
"Simple Harmonic Motion"
] | [
"Definition:Hypocycloid/Generator/Epicycle",
"Definition:Hypocycloid/Generator/Deferent",
"Definition:Tusi Couple",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Angular Velocity",
"Definition:Circle/Center",
... | [] |
proofwiki-11409 | Maximum Rate of Change of Y Coordinate of Astroid | Let $C_1$ and $C_2$ be the epicycle and deferent respectively of an astroid $H$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let the center $C$ of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$.
Let $P$ be the point on the circumference of $C_1$ w... | :400px
The rate of change of $\theta$ is given by:
:$\omega = \dfrac {\d \theta} {\d t}$
From Equation of Astroid: Parametric Form, the point $P = \tuple {x, y}$ is described by the parametric equation:
:$\begin {cases}
x & = a \cos^3 \theta \\
y & = a \sin^3 \theta
\end{cases}$
The rate of change of $y$ is given by:
{... | Let $C_1$ and $C_2$ be the [[Definition:Epicycle of Hypocycloid|epicycle]] and [[Definition:Deferent of Hypocycloid|deferent]] respectively of an [[Definition:Astroid|astroid]] $H$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:Center of Circle|center]] $O$ located at ... | :[[File:Astroid.png|400px]]
The [[Definition:Rate of Change with respect to Time|rate of change]] of $\theta$ is given by:
:$\omega = \dfrac {\d \theta} {\d t}$
From [[Equation of Astroid/Parametric Form|Equation of Astroid: Parametric Form]], the [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the [[... | Maximum Rate of Change of Y Coordinate of Astroid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_Y_Coordinate_of_Astroid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_Y_Coordinate_of_Astroid | [
"Astroids"
] | [
"Definition:Hypocycloid/Generator/Epicycle",
"Definition:Hypocycloid/Generator/Deferent",
"Definition:Astroid",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Angular Velocity",
"Definition:Circle/Center",
"D... | [
"File:Astroid.png",
"Definition:Rate of Change/Time",
"Equation of Astroid/Parametric Form",
"Definition:Point",
"Definition:Parametric Equation",
"Definition:Rate of Change/Time",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Derivative of Sine Function",
"Derivative of Compo... |
proofwiki-11410 | Equation of Deltoid | The point $P = \tuple {x, y}$ is described by the parametric equation:
:$\begin{cases}
x & = 2 b \cos \theta + b \cos 2 \theta \\
y & = 2 b \sin \theta - b \sin 2 \theta
\end{cases}$
where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$. | By definition, a deltoid is a hypocycloid with $3$ cusps.
:400px
By Equation of Hypocycloid, the equation of $H$ is given by:
:$\begin{cases}
x & = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta} \\
y & = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}
\end{cases}... | The [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the [[Definition:Parametric Equation|parametric equation]]:
:$\begin{cases}
x & = 2 b \cos \theta + b \cos 2 \theta \\
y & = 2 b \sin \theta - b \sin 2 \theta
\end{cases}$
where $\theta$ is the [[Definition:Angle|angle]] between the [[Definition:X-Axis|... | By definition, a [[Definition:Deltoid (Hypocycloid)|deltoid]] is a [[Definition:Hypocycloid|hypocycloid]] with $3$ [[Definition:Cusp of Hypocycloid|cusps]].
:[[File:Deltoid.png|400px]]
By [[Equation of Hypocycloid]], the equation of $H$ is given by:
:$\begin{cases}
x & = \paren {a - b} \cos \theta + b \map \cos {\pa... | Equation of Deltoid | https://proofwiki.org/wiki/Equation_of_Deltoid | https://proofwiki.org/wiki/Equation_of_Deltoid | [
"Deltoids (Hypocycloids)"
] | [
"Definition:Point",
"Definition:Parametric Equation",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center"
] | [
"Definition:Deltoid (Hypocycloid)",
"Definition:Hypocycloid",
"Definition:Hypocycloid/Cusp",
"File:Deltoid.png",
"Equation of Hypocycloid",
"Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii",
"Definition:Hypocycloid/Generator/Epicycle",
"Definition:Circle/Radius",
"Definition:Circl... |
proofwiki-11411 | Length of Arc of Deltoid | The total length of the arcs of a deltoid constructed within a deferent of radius $a$ is given by:
:$\LL = \dfrac {16 a} 3$ | Let $H$ be embedded in a cartesian plane with its center at the origin and one of its cusps positioned at $\tuple {a, 0}$.
:400px
We have that $\LL$ is $3$ times the length of one arc of the deltoid.
From Arc Length for Parametric Equations:
:$\ds \LL = 3 \int_{\theta \mathop = 0}^{\theta \mathop = 2 \pi/3} \sqrt {\par... | The total [[Definition:Arc Length|length of the arcs]] of a [[Definition:Deltoid (Hypocycloid)|deltoid]] constructed within a [[Definition:Deferent of Hypocycloid|deferent]] of [[Definition:Radius of Circle|radius]] $a$ is given by:
:$\LL = \dfrac {16 a} 3$ | Let $H$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its center at the [[Definition:Origin|origin]] and one of its [[Definition:Cusp of Hypocycloid|cusps]] positioned at $\tuple {a, 0}$.
:[[File:Deltoid.png|400px]]
We have that $\LL$ is $3$ times the [[Definition:Arc Length|length]] of one [... | Length of Arc of Deltoid | https://proofwiki.org/wiki/Length_of_Arc_of_Deltoid | https://proofwiki.org/wiki/Length_of_Arc_of_Deltoid | [
"Deltoids (Hypocycloids)"
] | [
"Definition:Arc Length",
"Definition:Deltoid (Hypocycloid)",
"Definition:Hypocycloid/Generator/Deferent",
"Definition:Circle/Radius"
] | [
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Hypocycloid/Cusp",
"File:Deltoid.png",
"Definition:Arc Length",
"Definition:Hypocycloid/Arc",
"Definition:Deltoid (Hypocycloid)",
"Arc Length for Parametric Equations",
"Equation of Deltoid",
"Sum of Squares of Sine a... |
proofwiki-11412 | Equation of Epicycloid | Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference of $C_1$.
Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_... | :600px
Let $C_1$ have rolled so that the line $OC$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
{{begin-eqn}}
{{eqn | l = x
| r = \paren {a + b} \co... | Let a [[Definition:Circle|circle]] $C_1$ of [[Definition:Radius of Circle|radius]] $b$ roll without slipping around the outside of a [[Definition:Circle|circle]] $C_2$ of [[Definition:Radius of Circle|radius]] $a$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:Center o... | :[[File:Epicycloid.png|600px]]
Let $C_1$ have rolled so that the [[Definition:Straight Line|line]] $OC$ through the [[Definition:Radius of Circle|radii]] of $C_1$ and $C_2$ is at [[Definition:Angle|angle]] $\theta$ to the [[Definition:X-Axis|$x$-axis]].
Let $C_1$ have turned through an angle $\phi$ to reach that poin... | Equation of Epicycloid | https://proofwiki.org/wiki/Equation_of_Epicycloid | https://proofwiki.org/wiki/Equation_of_Epicycloid | [
"Epicycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Tangent Circles",
"Definition:Poin... | [
"File:Epicycloid.png",
"Definition:Line/Straight Line",
"Definition:Circle/Radius",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Sine",
"Definition:Cosine",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Arc Length of Sector"
] |
proofwiki-11413 | Relation between Equations for Hypocycloid and Epicycloid | Consider the hypocycloid defined by the equations:
:$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$
:$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$
By replacing $b$ with $-b$, this converts to the equations which define an epicycloid:
:$x = \paren {a ... | {{begin-eqn}}
{{eqn | l = x
| r = \paren {a - \paren {-b} } \cos \theta + \paren {-b} \map \cos {\paren {\dfrac {a - \paren {-b} } {\paren {-b} } } \theta}
| c = putting $-b$ for $b$
}}
{{eqn | r = \paren {a + b} \cos \theta - b \map \cos {-\paren {\dfrac {a + b} b} \theta}
| c =
}}
{{eqn | r = \pare... | Consider the [[Definition:Hypocycloid|hypocycloid]] defined by the [[Equation of Hypocycloid|equations]]:
:$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$
:$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$
By replacing $b$ with $-b$, this converts to th... | {{begin-eqn}}
{{eqn | l = x
| r = \paren {a - \paren {-b} } \cos \theta + \paren {-b} \map \cos {\paren {\dfrac {a - \paren {-b} } {\paren {-b} } } \theta}
| c = putting $-b$ for $b$
}}
{{eqn | r = \paren {a + b} \cos \theta - b \map \cos {-\paren {\dfrac {a + b} b} \theta}
| c =
}}
{{eqn | r = \pare... | Relation between Equations for Hypocycloid and Epicycloid | https://proofwiki.org/wiki/Relation_between_Equations_for_Hypocycloid_and_Epicycloid | https://proofwiki.org/wiki/Relation_between_Equations_for_Hypocycloid_and_Epicycloid | [
"Hypocycloids",
"Epicycloids"
] | [
"Definition:Hypocycloid",
"Equation of Hypocycloid",
"Equation of Epicycloid",
"Definition:Epicycloid"
] | [
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-11414 | Equation of Nephroid | The point $P = \tuple {x, y}$ is described by the parametric equation:
:$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$
where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$. | By definition, a nephroid is an epicycloid with $2$ cusps.
:600px
By Equation of Epicycloid, the equation of $H$ is given by:
:$\begin{cases} x & = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta} \\ y & = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta} \end{cases}... | The [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the [[Definition:Parametric Equation|parametric equation]]:
:$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$
where $\theta$ is the [[Definition:Angle|angle]] between the [[Definition:X-Axis|... | By definition, a [[Definition:Nephroid|nephroid]] is an [[Definition:Epicycloid|epicycloid]] with $2$ [[Definition:Cusp of Hypocycloid|cusps]].
:[[File:Nephroid.png|600px]]
By [[Equation of Epicycloid]], the equation of $H$ is given by:
:$\begin{cases} x & = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {... | Equation of Nephroid | https://proofwiki.org/wiki/Equation_of_Nephroid | https://proofwiki.org/wiki/Equation_of_Nephroid | [
"Nephroids"
] | [
"Definition:Point",
"Definition:Parametric Equation",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center"
] | [
"Definition:Nephroid",
"Definition:Epicycloid",
"Definition:Hypocycloid/Cusp",
"File:Nephroid.png",
"Equation of Epicycloid"
] |
proofwiki-11415 | Length of Arc of Nephroid | The total length of the arcs of a nephroid constructed around a deferent of radius $a$ is given by:
:$\LL = 12 a$ | Let a nephroid $H$ be embedded in a cartesian plane with its center at the origin and its cusps positioned at $\tuple {\pm a, 0}$.
:600px
We have that $\LL$ is $2$ times the length of one arc of the nephroid.
From Arc Length for Parametric Equations:
:$\ds \LL = 2 \int_{\theta \mathop = 0}^{\theta \mathop = \pi} \sqrt ... | The total [[Definition:Arc Length|length of the arcs]] of a [[Definition:Nephroid|nephroid]] constructed around a [[Definition:Deferent of Epicycloid|deferent]] of [[Definition:Radius of Circle|radius]] $a$ is given by:
:$\LL = 12 a$ | Let a [[Definition:Nephroid|nephroid]] $H$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its center at the [[Definition:Origin|origin]] and its [[Definition:Cusp of Hypocycloid|cusps]] positioned at $\tuple {\pm a, 0}$.
:[[File:Nephroid.png|600px]]
We have that $\LL$ is $2$ times the [[Defini... | Length of Arc of Nephroid | https://proofwiki.org/wiki/Length_of_Arc_of_Nephroid | https://proofwiki.org/wiki/Length_of_Arc_of_Nephroid | [
"Nephroids"
] | [
"Definition:Arc Length",
"Definition:Nephroid",
"Definition:Epicycloid/Generator/Deferent",
"Definition:Circle/Radius"
] | [
"Definition:Nephroid",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Hypocycloid/Cusp",
"File:Nephroid.png",
"Definition:Arc Length",
"Definition:Epicycloid/Arc",
"Definition:Nephroid",
"Arc Length for Parametric Equations",
"Equation of Nephroid",
"Square of D... |
proofwiki-11416 | Supremum of Meet Image of Directed Set | Let $\struct {S, \preceq}$ be an up-complete meet semilattice.
Let $f: S \times S \to S$ be a mapping such that:
:$\forall \tuple {x, y} \in S \times S: \map f {x, y} = x \wedge y$
Let $D$ be directed subset of $S \times S$ in the simple order product $\struct {S \times S, \precsim}$ of $\struct {S, \preceq}$ and $\str... | By definition of image of set:
:$\map {f^\to} D = \set {x \wedge y: \tuple {x, y} \in D}$
By definition of subset:
:$\map {f^\to} D \subseteq \set {x \wedge y: x \in \map {\pr_1^\to} D, y \in \map {\pr_2^\to} D}$
By Up-Complete Product/Lemma 2:
:$D_1 := \map {\pr_1^\to} D$ is directed
and
:$D_2 := \map {\pr_2^\to} D$ i... | Let $\struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]].
Let $f: S \times S \to S$ be a [[Definition:Mapping|mapping]] such that:
:$\forall \tuple {x, y} \in S \times S: \map f {x, y} = x \wedge y$
Let $D$ be [[Definition:Directed Subset|directed subset... | By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\map {f^\to} D = \set {x \wedge y: \tuple {x, y} \in D}$
By definition of [[Definition:Subset|subset]]:
:$\map {f^\to} D \subseteq \set {x \wedge y: x \in \map {\pr_1^\to} D, y \in \map {\pr_2^\to} D}$
By [[Up-Complete Product/Lemma 2]]:
:$... | Supremum of Meet Image of Directed Set | https://proofwiki.org/wiki/Supremum_of_Meet_Image_of_Directed_Set | https://proofwiki.org/wiki/Supremum_of_Meet_Image_of_Directed_Set | [
"Up-Complete Semilattices"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Directed Subset",
"Definition:Simple Order Product",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Projection (Mapping Theory)/Second Projection",
"Definition:Image (Set Theory)/Mapping/... | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Up-Complete Product/Lemma 2",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Meet of Directed Subsets is Directed",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Meet is Inc... |
proofwiki-11417 | Meet is Increasing | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $f: S \times S \to S$ be a mapping such that:
:$\forall s, t \in S: \map f {s, t} = s \wedge t$
Then:
:$f$ is increasing as a mapping from the simple order product $\struct {S \times S, \precsim}$ of $\struct {S, \preceq}$ and $\struct {S, \preceq}$ into $\struct {S... | Let $\tuple {x, y}, \tuple {z, t} \in S \times S$ such that:
:$\tuple {x, y} \precsim \tuple {z, t}$
By definition of simple order product:
:$x \preceq z$ and $y \preceq t$
By Meet Semilattice is Ordered Structure:
:$x \wedge y \preceq z \wedge t$
By definition of $f$:
:$\map f {x, y} \preceq \map f {z, t}$
Thus by def... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $f: S \times S \to S$ be a [[Definition:Mapping|mapping]] such that:
:$\forall s, t \in S: \map f {s, t} = s \wedge t$
Then:
:$f$ is [[Definition:Increasing Mapping|increasing]] as a [[Definition:Mapping|mapping]] from the [[Definit... | Let $\tuple {x, y}, \tuple {z, t} \in S \times S$ such that:
:$\tuple {x, y} \precsim \tuple {z, t}$
By definition of [[Definition:Simple Order Product|simple order product]]:
:$x \preceq z$ and $y \preceq t$
By [[Meet Semilattice is Ordered Structure]]:
:$x \wedge y \preceq z \wedge t$
By definition of $f$:
:$\map ... | Meet is Increasing | https://proofwiki.org/wiki/Meet_is_Increasing | https://proofwiki.org/wiki/Meet_is_Increasing | [
"Meet Operation"
] | [
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Increasing/Mapping",
"Definition:Mapping",
"Definition:Simple Order Product"
] | [
"Definition:Simple Order Product",
"Meet Semilattice is Ordered Structure",
"Definition:Increasing/Mapping"
] |
proofwiki-11418 | Meet Preserves Directed Suprema/Lemma 2 | Let $x$ be an element of $S$, $D$ be a directed subset of $S$.
Then:
:$\paren {\sup D} \wedge x = \sup \set {d \wedge x: d \in D}$ | By Meet Precedes Operands:
:$\paren {\sup D} \wedge x \preceq \sup D$
By assumption:
:$\paren {\sup D} \wedge x \preceq \sup \set {\paren {\sup D} \wedge x \wedge d: d \in D}$
By definition of supremum:
:$\forall d \in D: d \preceq \sup D$
By Preceding iff Meet equals Less Operand:
:$\forall d \in D: d \wedge \sup D = ... | Let $x$ be an [[Definition:Element|element]] of $S$, $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Then:
:$\paren {\sup D} \wedge x = \sup \set {d \wedge x: d \in D}$ | By [[Meet Precedes Operands]]:
:$\paren {\sup D} \wedge x \preceq \sup D$
By assumption:
:$\paren {\sup D} \wedge x \preceq \sup \set {\paren {\sup D} \wedge x \wedge d: d \in D}$
By definition of [[Definition:Supremum of Set|supremum]]:
:$\forall d \in D: d \preceq \sup D$
By [[Preceding iff Meet equals Less Operan... | Meet Preserves Directed Suprema/Lemma 2 | https://proofwiki.org/wiki/Meet_Preserves_Directed_Suprema/Lemma_2 | https://proofwiki.org/wiki/Meet_Preserves_Directed_Suprema/Lemma_2 | [
"Up-Complete Semilattices"
] | [
"Definition:Element",
"Definition:Directed Subset"
] | [
"Meet Precedes Operands",
"Definition:Supremum of Set",
"Preceding iff Meet equals Less Operand",
"Meet is Associative",
"Meet is Commutative",
"Meet Semilattice is Ordered Structure",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Directed... |
proofwiki-11419 | Supremum by Suprema of Directed Set in Simple Order Product | Let $\struct {S, \preceq}$ be an up-complete meet semilattice.
Let $\struct {S \times S, \precsim}$ be the simple order product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$.
Let $D$ be a directed subset of $S \times S$.
Then:
:$\sup D = \tuple {\map \sup {\map {\pr_1^\to} D}, \map \sup {\map {\pr_2^\to} D} }$ | By Up-Complete Product:
:$\struct {S \times S, \precsim}$ is up-complete.
By definition of up-complete:
:$D$ admits a supremum.
By definition of Cartesian product:
:$\exists d_1, d_2 \in S: \sup D = \tuple {d_1, d_2}$
By Up-Complete Product/Lemma 2:
:$D_1 := \map {\pr_1^\to} D$ is directed
and
:$D_2 := \map {\pr_2^\to}... | Let $\struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]].
Let $\struct {S \times S, \precsim}$ be the [[Definition:Simple Order Product|simple order product]] of $\struct {S, \preceq}$ and $\struct {S, \preceq}$.
Let $D$ be a [[Definition:Directed Subset... | By [[Up-Complete Product]]:
:$\struct {S \times S, \precsim}$ is [[Definition:Up-Complete|up-complete]].
By definition of [[Definition:Up-Complete|up-complete]]:
:$D$ admits a [[Definition:Supremum of Set|supremum]].
By definition of [[Definition:Cartesian Product|Cartesian product]]:
:$\exists d_1, d_2 \in S: \sup D... | Supremum by Suprema of Directed Set in Simple Order Product | https://proofwiki.org/wiki/Supremum_by_Suprema_of_Directed_Set_in_Simple_Order_Product | https://proofwiki.org/wiki/Supremum_by_Suprema_of_Directed_Set_in_Simple_Order_Product | [
"Up-Complete Semilattices",
"Simple Order Product"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Simple Order Product",
"Definition:Directed Subset"
] | [
"Up-Complete Product",
"Definition:Up-Complete",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Definition:Cartesian Product",
"Up-Complete Product/Lemma 2",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Definition:... |
proofwiki-11420 | Evolute of Circle is its Center | The evolute of a circle is a single point: its center. | By definition, the evolute of $C$ is the locus of the centers of curvature of each point on $C$
{{WLOG}}, take the circle $C$ of radius $a$ whose center is positioned at the origin of a cartesian plane.
From Equation of Circle, $C$ has the equation:
:$x^2 + y^2 = a^2$
From the definition of curvature in cartesian form:... | The [[Definition:Evolute|evolute]] of a [[Definition:Circle|circle]] is a single [[Definition:Point|point]]: its [[Definition:Center of Circle|center]]. | By definition, the [[Definition:Evolute|evolute]] of $C$ is the [[Definition:Locus|locus]] of the [[Definition:Center of Curvature|centers of curvature]] of each [[Definition:Point|point]] on $C$
{{WLOG}}, take the [[Definition:Circle|circle]] $C$ of [[Definition:Radius of Circle|radius]] $a$ whose [[Definition:Center... | Evolute of Circle is its Center | https://proofwiki.org/wiki/Evolute_of_Circle_is_its_Center | https://proofwiki.org/wiki/Evolute_of_Circle_is_its_Center | [
"Examples of Evolutes",
"Circles"
] | [
"Definition:Evolute",
"Definition:Circle",
"Definition:Point",
"Definition:Circle/Center"
] | [
"Definition:Evolute",
"Definition:Locus",
"Definition:Center of Curvature",
"Definition:Point",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Equation of Circle",
"Definition:Curvature/Cartesian ... |
proofwiki-11421 | Radius of Curvature in Cartesian Form | Let $C$ be a curve defined by a real function which is twice differentiable.
Let $C$ be embedded in a cartesian plane.
The '''radius of curvature''' $\rho$ of $C$ at a point $P = \tuple {x, y}$ is given by:
:$\rho = \dfrac {\paren {1 + y'^2}^{3/2} } {\size {y' '} }$
where:
:$y' = \dfrac {\d y} {\d x}$ is the derivative... | By definition, the radius of curvature $\rho$ is given by:
:$\rho = \dfrac 1 {\size \kappa}$
where $\kappa$ is the curvature, given in Cartesian form as:
:$\kappa = \dfrac {y' '} {\paren {1 + y'^2}^{3/2} }$
As $\paren {1 + y'^2}^{3/2}$ is positive, it follows that:
:$\size {\dfrac {y' '} {\paren {1 + y'^2}^{3/2} } } = ... | Let $C$ be a [[Definition:Curve|curve]] defined by a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable Real Function|differentiable]].
Let $C$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]].
The '''[[Definition:Radius of Curvature|... | By definition, the [[Definition:Radius of Curvature|radius of curvature]] $\rho$ is given by:
:$\rho = \dfrac 1 {\size \kappa}$
where $\kappa$ is the [[Definition:Curvature|curvature]], given in [[Definition:Curvature/Cartesian Form|Cartesian form]] as:
:$\kappa = \dfrac {y' '} {\paren {1 + y'^2}^{3/2} }$
As $\paren {... | Radius of Curvature in Cartesian Form | https://proofwiki.org/wiki/Radius_of_Curvature_in_Cartesian_Form | https://proofwiki.org/wiki/Radius_of_Curvature_in_Cartesian_Form | [
"Radius of Curvature"
] | [
"Definition:Line/Curve",
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function",
"Definition:Cartesian Plane",
"Definition:Radius of Curvature",
"Definition:Point",
"Definition:Derivative",
"Definition:Derivative/Hi... | [
"Definition:Radius of Curvature",
"Definition:Curvature",
"Definition:Curvature/Cartesian Form",
"Definition:Positive/Real Number"
] |
proofwiki-11422 | Parametric Equations for Evolute/Formulation 1 | Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$.
The parametric equations for the evolute of $C$ can be expressed as:
:<nowiki>$\begin{cases}
X = x - \dfrac {y' \paren {1 + y'^2} } {y' '} \\
Y = y + \dfrac {1 + y'^2} {y''}
\end{cases}$</nowiki>
where:
:$\tuple {x, y}$ denotes the Cartesian ... | :400px
Let $P = \tuple {x, y}$ be a general point on $C$.
Let $Q = \tuple {X, Y}$ be the center of curvature of $C$ at $P$.
From the above diagram:
:$x - X = \pm \rho \sin \psi$
:$Y - y = \pm \rho \cos \psi$
where:
:$\rho$ is the radius of curvature of $C$ at $P$
:$\psi$ is the angle between the tangent to $C$ at $P$ a... | Let $C$ be a [[Definition:Curve|curve]] expressed as the [[Definition:Locus|locus]] of an [[Definition:Equation|equation]] $\map f {x, y} = 0$.
The [[Definition:Parametric Equation|parametric equations]] for the [[Definition:Evolute|evolute]] of $C$ can be expressed as:
:<nowiki>$\begin{cases}
X = x - \dfrac {y' \par... | :[[File:CenterOfCurvature.png|400px]]
Let $P = \tuple {x, y}$ be a general [[Definition:Point|point]] on $C$.
Let $Q = \tuple {X, Y}$ be the [[Definition:Center of Curvature|center of curvature]] of $C$ at $P$.
From the above diagram:
:$x - X = \pm \rho \sin \psi$
:$Y - y = \pm \rho \cos \psi$
where:
:$\rho$ is th... | Parametric Equations for Evolute/Formulation 1 | https://proofwiki.org/wiki/Parametric_Equations_for_Evolute/Formulation_1 | https://proofwiki.org/wiki/Parametric_Equations_for_Evolute/Formulation_1 | [
"Parametric Equations for Evolute"
] | [
"Definition:Line/Curve",
"Definition:Locus",
"Definition:Equation",
"Definition:Parametric Equation",
"Definition:Evolute",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Evolute",
"Definition:Derivative",
... | [
"File:CenterOfCurvature.png",
"Definition:Point",
"Definition:Center of Curvature",
"Definition:Radius of Curvature",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Radius of Curvature",
... |
proofwiki-11423 | Parametric Equations for Evolute/Formulation 2 | Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$.
The parametric equations for the evolute of $C$ can be expressed as:
:<nowiki>$\begin {cases}
X = x - \dfrac {y' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \\
Y = y + \dfrac {x' \paren {x'^2 + y'^2} } {x' y'' - y' x''}
\end {cases}$</nowiki>
wh... | :400px
Let $P = \tuple {x, y}$ be a general point on $C$.
Let $Q = \tuple {X, Y}$ be the center of curvature of $C$ at $P$.
From the above diagram:
{{begin-eqn}}
{{eqn | l = x - X
| r = \pm \rho \sin \psi
}}
{{eqn | l = Y - y
| r = \pm \rho \cos \psi
}}
{{end-eqn}}
where:
:$\rho$ is the radius of curvature ... | Let $C$ be a [[Definition:Curve|curve]] expressed as the [[Definition:Locus|locus]] of an [[Definition:Equation|equation]] $\map f {x, y} = 0$.
The [[Definition:Parametric Equation|parametric equations]] for the [[Definition:Evolute|evolute]] of $C$ can be expressed as:
:<nowiki>$\begin {cases}
X = x - \dfrac {y' \pa... | :[[File:CenterOfCurvature.png|400px]]
Let $P = \tuple {x, y}$ be a general [[Definition:Point|point]] on $C$.
Let $Q = \tuple {X, Y}$ be the [[Definition:Center of Curvature|center of curvature]] of $C$ at $P$.
From the above diagram:
{{begin-eqn}}
{{eqn | l = x - X
| r = \pm \rho \sin \psi
}}
{{eqn | l = Y... | Parametric Equations for Evolute/Formulation 2 | https://proofwiki.org/wiki/Parametric_Equations_for_Evolute/Formulation_2 | https://proofwiki.org/wiki/Parametric_Equations_for_Evolute/Formulation_2 | [
"Parametric Equations for Evolute"
] | [
"Definition:Line/Curve",
"Definition:Locus",
"Definition:Equation",
"Definition:Parametric Equation",
"Definition:Evolute",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Evolute",
"Definition:Derivative",
... | [
"File:CenterOfCurvature.png",
"Definition:Point",
"Definition:Center of Curvature",
"Definition:Radius of Curvature",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Radius of Curvature",
... |
proofwiki-11424 | Evolute of Parabola | The evolute of the parabola $y = x^2$ is the curve:
:$27 X^2 = 16 \paren {Y - \dfrac 1 2}^3$ | From Parametric Equations for Evolute: Formulation 1:
:<nowiki>$\begin {cases}
X = x - \dfrac {y' \paren {1 + y'^2} } {y' '} \\
Y = y + \dfrac {1 + y'^2} {y''}
\end{cases}$</nowiki>
where:
:$\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$
:$\tuple {X, Y}$ denotes the Cartesian coordinates of ... | The [[Definition:Evolute|evolute]] of the [[Definition:Parabola|parabola]] $y = x^2$ is the [[Definition:Curve|curve]]:
:$27 X^2 = 16 \paren {Y - \dfrac 1 2}^3$ | From [[Parametric Equations for Evolute/Formulation 1|Parametric Equations for Evolute: Formulation 1]]:
:<nowiki>$\begin {cases}
X = x - \dfrac {y' \paren {1 + y'^2} } {y' '} \\
Y = y + \dfrac {1 + y'^2} {y''}
\end{cases}$</nowiki>
where:
:$\tuple {x, y}$ denotes the [[Definition:Cartesian Coordinate System|Cartesia... | Evolute of Parabola | https://proofwiki.org/wiki/Evolute_of_Parabola | https://proofwiki.org/wiki/Evolute_of_Parabola | [
"Examples of Evolutes",
"Parabolas",
"Semicubical Parabola"
] | [
"Definition:Evolute",
"Definition:Parabola",
"Definition:Line/Curve"
] | [
"Parametric Equations for Evolute/Formulation 1",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Evolute",
"Definition:Derivative",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Parab... |
proofwiki-11425 | Normal to Curve is Tangent to Evolute | Let $C$ be a curve defined by a real function which is twice differentiable.
Let the curvature of $C$ be non-constant.
Let $P$ be a point on $C$.
Let $Q$ be the center of curvature of $C$ at $P$.
The normal to $C$ at $P$ is tangent to the evolute $E$ of $C$ at $Q$. | :400px
Let $P = \tuple {x, y}$ be a general point on $C$.
Let $Q = \tuple {X, Y}$ be the center of curvature of $C$ at $P$.
From the above diagram:
:$(1): \quad \begin{cases} x - X = \pm \rho \sin \psi \\ Y - y = \pm \rho \cos \psi \end{cases}$
where:
:$\rho$ is the radius of curvature of $C$ at $P$
:$\psi$ is the angl... | Let $C$ be a [[Definition:Curve|curve]] defined by a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable Real Function|differentiable]].
Let the [[Definition:Curvature|curvature]] of $C$ be non-[[Definition:Constant|constant]].
Let $P$ be a [[Definiti... | :[[File:CenterOfCurvature.png|400px]]
Let $P = \tuple {x, y}$ be a general [[Definition:Point|point]] on $C$.
Let $Q = \tuple {X, Y}$ be the [[Definition:Center of Curvature|center of curvature]] of $C$ at $P$.
From the above diagram:
:$(1): \quad \begin{cases} x - X = \pm \rho \sin \psi \\ Y - y = \pm \rho \cos \... | Normal to Curve is Tangent to Evolute | https://proofwiki.org/wiki/Normal_to_Curve_is_Tangent_to_Evolute | https://proofwiki.org/wiki/Normal_to_Curve_is_Tangent_to_Evolute | [
"Normals to Curves",
"Tangents",
"Evolutes"
] | [
"Definition:Line/Curve",
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function",
"Definition:Curvature",
"Definition:Constant",
"Definition:Point",
"Definition:Center of Curvature",
"Definition:Normal to Curve",
"... | [
"File:CenterOfCurvature.png",
"Definition:Point",
"Definition:Center of Curvature",
"Definition:Radius of Curvature",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Curvature",
"Definitio... |
proofwiki-11426 | Length of Arc of Evolute equals Difference in Radii of Curvature | Let $C$ be a curve defined by a real function which is twice differentiable.
Let the curvature of $C$ be non-constant.
The length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$. | :400px
Let $P = \tuple {x, y}$ be a general point on $C$.
Let $Q = \tuple {X, Y}$ be the center of curvature of $C$ at $P$.
From the above diagram:
:$(1): \quad \begin {cases} x - X = \pm \rho \sin \psi \\ Y - y = \pm \rho \cos \psi \end {cases}$
where:
:$\rho$ is the radius of curvature of $C$ at $P$
:$\psi$ is the an... | Let $C$ be a [[Definition:Curve|curve]] defined by a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable Real Function|differentiable]].
Let the [[Definition:Curvature|curvature]] of $C$ be non-[[Definition:Constant|constant]].
The [[Definition:Arc Le... | :[[File:CenterOfCurvature.png|400px]]
Let $P = \tuple {x, y}$ be a general [[Definition:Point|point]] on $C$.
Let $Q = \tuple {X, Y}$ be the [[Definition:Center of Curvature|center of curvature]] of $C$ at $P$.
From the above diagram:
:$(1): \quad \begin {cases} x - X = \pm \rho \sin \psi \\ Y - y = \pm \rho \cos ... | Length of Arc of Evolute equals Difference in Radii of Curvature | https://proofwiki.org/wiki/Length_of_Arc_of_Evolute_equals_Difference_in_Radii_of_Curvature | https://proofwiki.org/wiki/Length_of_Arc_of_Evolute_equals_Difference_in_Radii_of_Curvature | [
"Evolutes"
] | [
"Definition:Line/Curve",
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function",
"Definition:Curvature",
"Definition:Constant",
"Definition:Arc Length",
"Definition:Evolute",
"Definition:Point",
"Definition:Radius... | [
"File:CenterOfCurvature.png",
"Definition:Point",
"Definition:Center of Curvature",
"Definition:Radius of Curvature",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Curvature",
"Definitio... |
proofwiki-11427 | Curve is Involute of Evolute | Let $C$ be a curve defined by a real function which is twice differentiable.
Let the curvature of $C$ be non-constant.
Let $E$ be the evolute $C$.
Then the involute of $E$ is $C$. | From Length of Arc of Evolute equals Difference in Radii of Curvature:
:the length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$.
Thus $C$ exhibits precisely the property of the ... | Let $C$ be a [[Definition:Curve|curve]] defined by a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable Real Function|differentiable]].
Let the [[Definition:Curvature|curvature]] of $C$ be non-[[Definition:Constant|constant]].
Let $E$ be the [[Defini... | From [[Length of Arc of Evolute equals Difference in Radii of Curvature]]:
:the [[Definition:Arc Length|length of arc]] of the [[Definition:Evolute|evolute]] $E$ of $C$ between any two [[Definition:Point|points]] $Q_1$ and $Q_2$ of $C$ is equal to the difference between the [[Definition:Radius of Curvature|radii of cu... | Curve is Involute of Evolute | https://proofwiki.org/wiki/Curve_is_Involute_of_Evolute | https://proofwiki.org/wiki/Curve_is_Involute_of_Evolute | [
"Evolutes",
"Involutes"
] | [
"Definition:Line/Curve",
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function",
"Definition:Curvature",
"Definition:Constant",
"Definition:Evolute",
"Definition:Involute"
] | [
"Length of Arc of Evolute equals Difference in Radii of Curvature",
"Definition:Arc Length",
"Definition:Evolute",
"Definition:Point",
"Definition:Radius of Curvature",
"Definition:Point",
"Definition:Involute"
] |
proofwiki-11428 | Evolute of Cycloid is Cycloid | The evolute of a cycloid is another cycloid. | Let $C$ be the cycloid defined by the equations:
:$\begin {cases} x = a \paren {\theta - \sin \theta} \\ y = a \paren {1 - \cos \theta} \end {cases}$
From Parametric Equations for Evolute: Formulation 2:
:$(1): \quad \begin {cases} X = x - \dfrac {y' \paren {x'^2 + y'^2} } {x' y' ' - y' x' '} \\ Y = y + \dfrac {x' \par... | The [[Definition:Evolute|evolute]] of a [[Definition:Cycloid|cycloid]] is another [[Definition:Cycloid|cycloid]]. | Let $C$ be the [[Definition:Cycloid|cycloid]] defined by the equations:
:$\begin {cases} x = a \paren {\theta - \sin \theta} \\ y = a \paren {1 - \cos \theta} \end {cases}$
From [[Parametric Equations for Evolute/Formulation 2|Parametric Equations for Evolute: Formulation 2]]:
:$(1): \quad \begin {cases} X = x - \df... | Evolute of Cycloid is Cycloid | https://proofwiki.org/wiki/Evolute_of_Cycloid_is_Cycloid | https://proofwiki.org/wiki/Evolute_of_Cycloid_is_Cycloid | [
"Examples of Evolutes",
"Cycloids"
] | [
"Definition:Evolute",
"Definition:Cycloid",
"Definition:Cycloid"
] | [
"Definition:Cycloid",
"Parametric Equations for Evolute/Formulation 2",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Evolute",
"Definition:Derivative",
"Definition:Derivative/Higher Derivatives/Second Derivati... |
proofwiki-11429 | Meet of Directed Subsets is Directed | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $D_1, D_2$ be directed subset of $S$.
Then:
:$\set {x \wedge y: x \in D_1, y \in D_2}$ is a directed subset of $S$. | Let $a, b \in \set {x \wedge y: x \in D_1, y \in D_2}$.
Then:
:$\exists x \in D_1, y \in D_2: a = x \wedge y$
and
:$\exists z \in D_1, t \in D_2: b = z \wedge t$
By definition of directed subset:
:$\exists g \in D_1: x \preceq g \land z \preceq g$
and
:$\exists h \in D_2: y \preceq h \land t \preceq h$
By Meet Semilatt... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $D_1, D_2$ be [[Definition:Directed Subset|directed subset]] of $S$.
Then:
:$\set {x \wedge y: x \in D_1, y \in D_2}$ is a [[Definition:Directed Subset|directed subset]] of $S$. | Let $a, b \in \set {x \wedge y: x \in D_1, y \in D_2}$.
Then:
:$\exists x \in D_1, y \in D_2: a = x \wedge y$
and
:$\exists z \in D_1, t \in D_2: b = z \wedge t$
By definition of [[Definition:Directed Subset|directed subset]]:
:$\exists g \in D_1: x \preceq g \land z \preceq g$
and
:$\exists h \in D_2: y \preceq h \l... | Meet of Directed Subsets is Directed | https://proofwiki.org/wiki/Meet_of_Directed_Subsets_is_Directed | https://proofwiki.org/wiki/Meet_of_Directed_Subsets_is_Directed | [
"Meet Semilattices",
"Directed Preorderings"
] | [
"Definition:Meet Semilattice",
"Definition:Directed Subset",
"Definition:Directed Subset"
] | [
"Definition:Directed Subset",
"Meet Semilattice is Ordered Structure",
"Definition:Directed Subset"
] |
proofwiki-11430 | Image of Directed Subset under Increasing Mapping is Directed | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.
Let $f: S \to T$ be an increasing mapping.
Let $D$ be a directed subset of $S$.
Then
:$f^\to \left({D}\right)$ is also a directed subset of $T$
where
:$f^\to \left({D}\right)$ denotes the image of $D$ under $f$. | Let $x, y \in f^\to\left({D}\right)$.
By definition of image of set:
:$\exists a \in D: x = f \left({a}\right)$
and
:$\exists b \in D: y = f \left({b}\right)$
By definition of directed subset:
:$\exists c \in D: a \preceq c \land b \preceq c$
By definition of image of set:
:$f\left({c}\right) \in f^\to\left({D}\right)$... | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Then
:$f^\to \left({D}\right)$ is also a [[Definition:Directed Su... | Let $x, y \in f^\to\left({D}\right)$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists a \in D: x = f \left({a}\right)$
and
:$\exists b \in D: y = f \left({b}\right)$
By definition of [[Definition:Directed Subset|directed subset]]:
:$\exists c \in D: a \preceq c \land b \preceq c$... | Image of Directed Subset under Increasing Mapping is Directed | https://proofwiki.org/wiki/Image_of_Directed_Subset_under_Increasing_Mapping_is_Directed | https://proofwiki.org/wiki/Image_of_Directed_Subset_under_Increasing_Mapping_is_Directed | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Directed Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Increasing/Mapping",
"Definition:Directed Subset"
] |
proofwiki-11431 | Mass of Sun from Universal Gravitational Constant | Let the universal gravitational constant be known.
Let the mean distance from the Earth to the sun be known.
Then it is possible to calculate the mass of the sun. | From Kepler's Third Law of Planetary Motion:
:$T^2 = \paren {\dfrac {4 \pi^2} {G M} } a^3$
where:
:$T$ is the orbital period of the planet in question (in this case, the Earth)
:$a$ is the distance from the planet (in this case, the Earth) to the sun
:$M$ is the mass of the sun
:$G$ is the universal gravitational const... | Let the [[Definition:Universal Gravitational Constant|universal gravitational constant]] be known.
Let the [[Definition:Arithmetic Mean|mean]] [[Definition:Distance between Points|distance]] from the [[Definition:Earth|Earth]] to the [[Definition:Sun|sun]] be known.
Then it is possible to calculate the [[Definition:... | From [[Kepler's Third Law of Planetary Motion]]:
:$T^2 = \paren {\dfrac {4 \pi^2} {G M} } a^3$
where:
:$T$ is the [[Definition:Orbital Period|orbital period]] of the [[Definition:Planet|planet]] in question (in this case, the [[Definition:Earth|Earth]])
:$a$ is the [[Definition:Distance between Points|distance]] from t... | Mass of Sun from Universal Gravitational Constant | https://proofwiki.org/wiki/Mass_of_Sun_from_Universal_Gravitational_Constant | https://proofwiki.org/wiki/Mass_of_Sun_from_Universal_Gravitational_Constant | [
"Universal Gravitational Constant",
"Celestial Mechanics"
] | [
"Definition:Universal Gravitational Constant",
"Definition:Arithmetic Mean",
"Definition:Distance between Points",
"Definition:Earth",
"Definition:Sun",
"Definition:Mass",
"Definition:Sun"
] | [
"Kepler's Laws of Planetary Motion/Third Law",
"Definition:Orbit (Physics)/Period",
"Definition:Planet",
"Definition:Earth",
"Definition:Distance between Points",
"Definition:Planet",
"Definition:Earth",
"Definition:Sun",
"Definition:Mass",
"Definition:Sun",
"Definition:Universal Gravitational C... |
proofwiki-11432 | Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets | Let $\struct {S, \preceq}$ be an up-complete meet semilattice.
Let $\struct {S \times S, \precsim}$ be the simple order product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$.
Let $f: S \times S \to S$ be a mapping such that:
:$\forall s, t \in S: \map f {s, t} = s \wedge t$
and:
:$f$ preserves directed suprema.
... | By Up-Complete Product:
:$\struct {S \times S, \precsim}$ is up-complete.
By Up-Complete Product/Lemma 1:
:$D_1 \times D_2$ is directed subsets of $S \times S$
By definition of mapping preserves directed suprema:
:$f$ preserves the supremum of $D_1 \times D_2$
By definition of up-complete:
:$D_1 \times D_2$ admits a su... | Let $\struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]].
Let $\struct {S \times S, \precsim}$ be the [[Definition:Simple Order Product|simple order product]] of $\struct {S, \preceq}$ and $\struct {S, \preceq}$.
Let $f: S \times S \to S$ be a [[Definiti... | By [[Up-Complete Product]]:
:$\struct {S \times S, \precsim}$ is [[Definition:Up-Complete|up-complete]].
By [[Up-Complete Product/Lemma 1]]:
:$D_1 \times D_2$ is [[Definition:Directed Subset|directed subsets]] of $S \times S$
By definition of [[Definition:Mapping Preserves Directed Supremum|mapping preserves directed... | Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets | https://proofwiki.org/wiki/Meet_is_Directed_Suprema_Preserving_implies_Meet_of_Suprema_equals_Supremum_of_Meet_of_Directed_Subsets | https://proofwiki.org/wiki/Meet_is_Directed_Suprema_Preserving_implies_Meet_of_Suprema_equals_Supremum_of_Meet_of_Directed_Subsets | [
"Up-Complete Semilattices"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Simple Order Product",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Directed Subset"
] | [
"Up-Complete Product",
"Definition:Up-Complete",
"Up-Complete Product/Lemma 1",
"Definition:Directed Subset",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Definition:Supremum",
"Supremum o... |
proofwiki-11433 | Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice.
Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that:
:$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$
where
:$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$
Let
:$\f... | Let $I, J \in \map {\it Ids} {\mathscr S}$ such that
:$\set {I, J}$ admits an infimum in $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$.
By definition of image of set:
:$\map {f^\to} {\set {I, J} } = \set {\map f I, \map f J}$
Thus by definition of meet semilattice:
:$\map {f^\to} {\set {I, J} }$ admits an infimum ... | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]].
Let $f: \map {\it Ids} {\mathscr S} \to S$ be a [[Definition:Mapping|mapping]] such that:
:$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$
where
:$... | Let $I, J \in \map {\it Ids} {\mathscr S}$ such that
:$\set {I, J}$ admits an [[Definition:Infimum of Set|infimum]] in $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\map {f^\to} {\set {I, J} } = \set {\map f I, \map f J}$
Thus by defi... | Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving | https://proofwiki.org/wiki/Meet_of_Suprema_equals_Supremum_of_Meet_of_Ideals_implies_Ideal_Supremum_is_Meet_Preserving | https://proofwiki.org/wiki/Meet_of_Suprema_equals_Supremum_of_Meet_of_Ideals_implies_Ideal_Supremum_is_Meet_Preserving | [
"Up-Complete Semilattices"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set",
"Definition:Mapping Preserves Infimum/Meet",
"Definition:Mapping"
] | [
"Definition:Infimum of Set",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Meet Semilattice",
"Definition:Infimum of Set",
"Meet in Set of Ideals"
] |
proofwiki-11434 | Quaternion Multplication is not Commutative | The operation of multplication on the quaternions $H$ is not commutative. | By definition of multplication:
{{begin-eqn}}
{{eqn | l = \mathbf i \times \mathbf j
| r = \mathbf k
| c =
}}
{{eqn | l = \mathbf j \times \mathbf i
| r = -\mathbf k
| c =
}}
{{end-eqn}}
{{qed}} | The operation of [[Definition:Quaternion Multiplication|multplication]] on the [[Definition:Quaternion|quaternions]] $H$ is not [[Definition:Commutative Operation|commutative]]. | By definition of [[Definition:Quaternion Multiplication|multplication]]:
{{begin-eqn}}
{{eqn | l = \mathbf i \times \mathbf j
| r = \mathbf k
| c =
}}
{{eqn | l = \mathbf j \times \mathbf i
| r = -\mathbf k
| c =
}}
{{end-eqn}}
{{qed}} | Quaternion Multplication is not Commutative | https://proofwiki.org/wiki/Quaternion_Multplication_is_not_Commutative | https://proofwiki.org/wiki/Quaternion_Multplication_is_not_Commutative | [
"Quaternions"
] | [
"Definition:Quaternion/Multiplication",
"Definition:Quaternion",
"Definition:Commutative/Operation"
] | [
"Definition:Quaternion/Multiplication"
] |
proofwiki-11435 | Field Norm of Quaternion is not Norm | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
The field norm of $\mathbf x$:
:$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$
is not a norm in the abstract algebraic context of a division ring. | Each of the norm axioms is examined in turn: | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\overline {\mathbf x}$ be the [[Definition:Conjugate Quaternion|conjugate]] of $\mathbf x$.
The [[Definition:Field Norm of Quaternion|field norm]] of $\mathbf x$:
:$\map n {\mathbf x} := \size {\ma... | Each of the [[Axiom:Multiplicative Norm Axioms|norm axioms]] is examined in turn: | Field Norm of Quaternion is not Norm | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_not_Norm | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_not_Norm | [
"Quaternions",
"Norm Theory"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Field Norm of Quaternion",
"Definition:Norm/Division Ring",
"Definition:Abstract Algebra",
"Definition:Division Ring"
] | [
"Axiom:Multiplicative Norm Axioms",
"Axiom:Multiplicative Norm Axioms"
] |
proofwiki-11436 | Field Norm of Quaternion is Positive Definite | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
The field norm of $\mathbf x$:
:$\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$
is positive definite. | {{begin-eqn}}
{{eqn | l = \map n {\mathbf x}
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \cmod {\mathbf x \overline {\mathbf x} }
| r = 0
| c = {{Defof|Field Norm of Quaternion}}
}}
{{eqn | ll= \leadstoandfrom
| l = a^2 + b^2 + c^2 + d^2
| r = 0
| c =
}}
{{eqn | ... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\overline {\mathbf x}$ be the [[Definition:Conjugate Quaternion|conjugate]] of $\mathbf x$.
The [[Definition:Field Norm of Quaternion|field norm]] of $\mathbf x$:
:$\map n {\mathbf x} := \cmod {\ma... | {{begin-eqn}}
{{eqn | l = \map n {\mathbf x}
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \cmod {\mathbf x \overline {\mathbf x} }
| r = 0
| c = {{Defof|Field Norm of Quaternion}}
}}
{{eqn | ll= \leadstoandfrom
| l = a^2 + b^2 + c^2 + d^2
| r = 0
| c =
}}
{{eqn | ... | Field Norm of Quaternion is Positive Definite | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_Positive_Definite | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_Positive_Definite | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Field Norm of Quaternion",
"Definition:Positive Definite (Ring)"
] | [
"Definition:Positive Definite (Ring)"
] |
proofwiki-11437 | Field Norm of Quaternion is Multiplicative | Let $\mathbf x$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
The field norm of $\mathbf x$:
:$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$
is a multiplicative function. | {{begin-eqn}}
{{eqn | l = \map n {\mathbf x \mathbf y}
| r = \mathbf x \, \mathbf y \ \overline {\mathbf x \, \mathbf y}
| c = {{Defof|Field Norm of Quaternion}}
}}
{{eqn | r = \mathbf x \, \mathbf y \paren {\overline {\mathbf y} \, \overline {\mathbf x} }
| c = Product of Quaternion Conjugates
}}
{{e... | Let $\mathbf x$ be a [[Definition:Quaternion|quaternion]].
Let $\overline {\mathbf x}$ be the [[Definition:Conjugate Quaternion|conjugate]] of $\mathbf x$.
The [[Definition:Field Norm of Quaternion|field norm]] of $\mathbf x$:
:$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$
is a [[Definition:Multi... | {{begin-eqn}}
{{eqn | l = \map n {\mathbf x \mathbf y}
| r = \mathbf x \, \mathbf y \ \overline {\mathbf x \, \mathbf y}
| c = {{Defof|Field Norm of Quaternion}}
}}
{{eqn | r = \mathbf x \, \mathbf y \paren {\overline {\mathbf y} \, \overline {\mathbf x} }
| c = [[Product of Quaternion Conjugates]]
}}... | Field Norm of Quaternion is Multiplicative | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_Multiplicative | https://proofwiki.org/wiki/Field_Norm_of_Quaternion_is_Multiplicative | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Field Norm of Quaternion",
"Definition:Multiplicative Arithmetic Function"
] | [
"Product of Quaternion Conjugates"
] |
proofwiki-11438 | Equivalence of Definitions of Oscillation of Real Function at Point | Let $X$ and $Y$ be real sets.
Let $f: X \to Y$ be a real function.
Let $x \in X$.
{{TFAE|def = Oscillation of Real Function at Point}} | === Definitions 1 and 2 are equivalent ===
We reformulate Definition 1 into Definition 1' by:
:substituting the definition of $\map {\omega_f} {U \cap X}$ into the definition of $\map {\omega_f} x$
Definition 1':
:$\ds \map {\omega_f} x := \inf_{U \mathop \in \NN_x} \paren {\sup_{y, z \mathop \in U \cap X} \size {\map ... | Let $X$ and $Y$ be [[Definition:Real Number|real]] [[Definition:Set|sets]].
Let $f: X \to Y$ be a [[Definition:Real Function|real function]].
Let $x \in X$.
{{TFAE|def = Oscillation of Real Function at Point}} | === Definitions 1 and 2 are equivalent ===
We reformulate [[Definition:Oscillation of Real Function at Point/Infimum|Definition 1]] into Definition 1' by:
:substituting the definition of $\map {\omega_f} {U \cap X}$ into the definition of $\map {\omega_f} x$
Definition 1':
:$\ds \map {\omega_f} x := \inf_{U \mathop \... | Equivalence of Definitions of Oscillation of Real Function at Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Oscillation_of_Real_Function_at_Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Oscillation_of_Real_Function_at_Point | [
"Oscillation"
] | [
"Definition:Real Number",
"Definition:Set",
"Definition:Real Function",
"Definition:Set"
] | [
"Definition:Oscillation/Real Space/Oscillation at Point/Infimum",
"Definition:Oscillation/Real Space/Oscillation at Point/Epsilon",
"Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)",
"Definition:Oscillation/Real Space/Oscillation at Point",
"Definition:Real Function",
"De... |
proofwiki-11439 | Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars | It is not possible to extend the complex numbers to an algebra of $3$ dimensions with real scalars. | {{AimForCont}} that $\set {1, i, j}$ forms a basis for an algebra of $3$ dimensions with real scalars.
Let $1$ and $i$ have their usual properties as they do as complex numbers:
:$\forall a: 1 a = a 1 = a$
:$i \cdot i = -1$
Then:
:$i j = a_1 + a_2 i + a_3 j$
for some $a_1, a_2, a_3 \in \R$.
Multiplying through by $i$:
... | It is not possible to extend the [[Definition:Complex Number|complex numbers]] to an [[Definition:Algebra over Field|algebra]] of $3$ [[Definition:Dimension (Linear Algebra)|dimensions]] with [[Definition:Real Number|real]] [[Definition:Scalar (Vector Space)|scalars]]. | {{AimForCont}} that $\set {1, i, j}$ forms a [[Definition:Basis (Linear Algebra)|basis]] for an [[Definition:Algebra over Field|algebra]] of $3$ [[Definition:Dimension (Linear Algebra)|dimensions]] with [[Definition:Real Number|real]] [[Definition:Scalar (Vector Space)|scalars]].
Let $1$ and $i$ have their usual prope... | Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Extended_to_Algebra_in_Three_Dimensions_with_Real_Scalars | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Extended_to_Algebra_in_Three_Dimensions_with_Real_Scalars | [
"Algebras"
] | [
"Definition:Complex Number",
"Definition:Algebra over Field",
"Definition:Dimension (Linear Algebra)",
"Definition:Real Number",
"Definition:Scalar/Vector Space"
] | [
"Definition:Basis (Linear Algebra)",
"Definition:Algebra over Field",
"Definition:Dimension (Linear Algebra)",
"Definition:Real Number",
"Definition:Scalar/Vector Space",
"Definition:Complex Number",
"Proof by Contradiction"
] |
proofwiki-11440 | Supremum of Simple Order Product | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets.
Let $\struct {S_1 \times S_2, \precsim}$ be the simple order product of $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$.
Let $X_1$ be a non-empty subset of $S_1$, $X_2$ be a non-empty subset of $S_2$ such that
:$X_1$ and $X_2$ admi... | We will prove that:
:$\tuple {\sup X_1, \sup X_2}$ is upper bound for $X_1 \times X_2$
Let $\tuple {a, b} \in X_1 \times X_2$.
By definition of Cartesian product:
:$a \in X_1$ and $b \in X_2$
By definitions of supremum and upper bound:
:$a \preceq_1 \sup X_1$ and $b \preceq_2 \sup X_2$
Thus by definition of simple orde... | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S_1 \times S_2, \precsim}$ be the [[Definition:Simple Order Product|simple order product]] of $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$.
Let $X_1$ be a [[Definition:Non-Empty Set|non... | We will prove that:
:$\tuple {\sup X_1, \sup X_2}$ is [[Definition:Upper Bound of Set|upper bound]] for $X_1 \times X_2$
Let $\tuple {a, b} \in X_1 \times X_2$.
By definition of [[Definition:Cartesian Product|Cartesian product]]:
:$a \in X_1$ and $b \in X_2$
By definitions of [[Definition:Supremum of Set|supremum]] ... | Supremum of Simple Order Product | https://proofwiki.org/wiki/Supremum_of_Simple_Order_Product | https://proofwiki.org/wiki/Supremum_of_Simple_Order_Product | [
"Simple Order Product",
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Simple Order Product",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Cartesian Product",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Simple Order Product",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Non-Empty Set",
"Definit... |
proofwiki-11441 | Vectors in Three Dimensional Space with Cross Product forms Lie Algebra | Let $S$ be the set of vectors in $3$ dimensional Euclidean space.
Let $\times$ denote the vector cross product on $S$.
Then $\struct {S, \times}$ is a Lie algebra. | By definition of Lie algebra, it suffices to prove two properties:
:$(1): \forall a \in S: a \times a = 0$
:$(2): \forall a, b, c \in S: a \times \paren {b \times c} + b \times \paren {c \times a} + c \times \paren {a \times b} = 0$ | Let $S$ be the [[Definition:Set|set]] of [[Definition:Vector (Linear Algebra)|vectors]] in $3$ [[Definition:Dimension (Linear Algebra)|dimensional]] [[Definition:Euclidean Space|Euclidean space]].
Let $\times$ denote the [[Definition:Vector Cross Product|vector cross product]] on $S$.
Then $\struct {S, \times}$ is a... | By definition of [[Definition:Lie Algebra|Lie algebra]], it suffices to prove two properties:
:$(1): \forall a \in S: a \times a = 0$
:$(2): \forall a, b, c \in S: a \times \paren {b \times c} + b \times \paren {c \times a} + c \times \paren {a \times b} = 0$ | Vectors in Three Dimensional Space with Cross Product forms Lie Algebra | https://proofwiki.org/wiki/Vectors_in_Three_Dimensional_Space_with_Cross_Product_forms_Lie_Algebra | https://proofwiki.org/wiki/Vectors_in_Three_Dimensional_Space_with_Cross_Product_forms_Lie_Algebra | [
"Lie Algebras",
"Vector Cross Product"
] | [
"Definition:Set",
"Definition:Vector/Linear Algebra",
"Definition:Dimension (Linear Algebra)",
"Definition:Euclidean Space",
"Definition:Vector Cross Product",
"Definition:Lie Algebra"
] | [
"Definition:Lie Algebra"
] |
proofwiki-11442 | Bott-Milnor-Kervaire 1,2,4,8 Theorem | Let $A$ be a division algebra with real scalars.
Then the dimension of $A$ is either:
:$1$: the real numbers $\R$
:$2$: the complex numbers $\C$
:$4$: the quaternions $\Bbb H$
or:
:$8$: the octonions $\Bbb O$. | {{ProofWanted}}
{{Namedfor|Raoul Bott|name2 = John Willard Milnor|name3 = Michel André Kervaire|cat = Bott|cat2 = Milnor|cat3 = Kervaire}} | Let $A$ be a [[Definition:Division Algebra|division algebra]] with [[Definition:Real Number|real]] [[Definition:Scalar (Vector Space)|scalars]].
Then the [[Definition:Dimension (Linear Algebra)|dimension]] of $A$ is either:
:$1$: the [[Definition:Real Number|real numbers]] $\R$
:$2$: the [[Definition:Complex Number|c... | {{ProofWanted}}
{{Namedfor|Raoul Bott|name2 = John Willard Milnor|name3 = Michel André Kervaire|cat = Bott|cat2 = Milnor|cat3 = Kervaire}} | Bott-Milnor-Kervaire 1,2,4,8 Theorem | https://proofwiki.org/wiki/Bott-Milnor-Kervaire_1,2,4,8_Theorem | https://proofwiki.org/wiki/Bott-Milnor-Kervaire_1,2,4,8_Theorem | [
"Division Algebras"
] | [
"Definition:Division Algebra",
"Definition:Real Number",
"Definition:Scalar/Vector Space",
"Definition:Dimension (Linear Algebra)",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Quaternion",
"Definition:Octonion"
] | [] |
proofwiki-11443 | Montel's Theorem | Let $U \subseteq \C$ be an open subset of the complex numbers.
Let $\map \HH U$ be the space of holomorphic mappings on $U$.
Then a family of mappings $\FF \subseteq \map \HH U$ is normal {{iff}} $\FF$ is locally bounded. | === Normal implies locally bounded ===
By the Arzelà-Ascoli Theorem, every normal family is locally bounded. | Let $U \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open subset]] of the [[Definition:Complex Number|complex numbers]].
Let $\map \HH U$ be the space of [[Definition:Holomorphic Function|holomorphic mappings]] on $U$.
Then a [[Definition:Indexed Family|family]] of [[Definition:Mapping|mappings]] $\FF... | === Normal implies locally bounded ===
By the [[Arzelà-Ascoli Theorem]], every [[Definition:Normal Family|normal family]] is [[Definition:Locally Bounded Family of Mappings|locally bounded]]. | Montel's Theorem | https://proofwiki.org/wiki/Montel's_Theorem | https://proofwiki.org/wiki/Montel's_Theorem | [
"Complex Analysis"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Complex Number",
"Definition:Holomorphic Function",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Normal Family",
"Definition:Locally Bounded/Family of Mappings"
] | [
"Arzelà-Ascoli Theorem",
"Definition:Normal Family",
"Definition:Locally Bounded/Family of Mappings",
"Arzelà-Ascoli Theorem",
"Definition:Locally Bounded/Family of Mappings",
"Definition:Normal Family",
"Definition:Locally Bounded/Family of Mappings"
] |
proofwiki-11444 | Vitali's Convergence Theorem | Let $U$ be an open, connected subset of $\C$.
Let $S \subseteq U$ contain a limit point $\sigma$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a normal family of holomorphic mappings $f_n : U \to \C$.
Let $\sequence {f_n}_{n \mathop \in \N}$ converge to some holomorphic mapping $f : U \to \C$ at $\sigma$.
Then $f_n$... | {{AimForCont}} there exists some compact subset $K$ of $U$ such that $f_n$ does not converge uniformly to $f$ on $K$.
Consider $K^* := K \cup \set \sigma$.
From Subsets Inherit Uniform Convergence, $f_n$ does not converge uniformly to $f$ on $K^*$.
From Uniformly Convergent iff Difference Under Supremum Norm Vanishes, ... | Let $U$ be an [[Definition: Open Set of Metric Space | open]], [[Definition: Connected | connected]] subset of $\C$.
Let $S \subseteq U$ contain a [[Definition:Limit Point (Complex Analysis)|limit point]] $\sigma$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition: Normal Family | normal family]] of [[Defini... | {{AimForCont}} there exists some compact subset $K$ of $U$ such that $f_n$ does not converge uniformly to $f$ on $K$.
Consider $K^* := K \cup \set \sigma$.
From [[Subsets Inherit Uniform Convergence]], $f_n$ does not converge uniformly to $f$ on $K^*$.
From [[Uniformly Convergent iff Difference Under Supremum Norm V... | Vitali's Convergence Theorem | https://proofwiki.org/wiki/Vitali's_Convergence_Theorem | https://proofwiki.org/wiki/Vitali's_Convergence_Theorem | [
"Complex Analysis"
] | [
"Definition: Open Set of Metric Space ",
"Definition: Connected ",
"Definition:Limit Point/Complex Analysis",
"Definition: Normal Family ",
"Definition:Analytic Function/Complex Plane",
"Definition:Mapping",
"Definition:Pointwise Convergence ",
"Definition:Holomorphic Function/Complex Plane",
"Defin... | [
"Uniform Convergence is Hereditary",
"Uniformly Convergent iff Difference Under Supremum Metric Vanishes",
"Definition:Logical Equivalence",
"Definition:Supremum Norm",
"Finite Union of Compact Sets is Compact",
"Definition:Compact Space/Metric Space",
"Definition: Normal Family",
"Definition:Subseque... |
proofwiki-11445 | Identity Theorem | Let $U$ be an open connected subset of the complex plane $\C$.
Let $f$ and $g$ be complex functions whose domain is $U$.
Let $S = \set {z \in U: \map f z = \map g z}$.
Let $f$ and $g$ be analytic on $U$.
Let $S$ have a limit point in $U$.
Then:
:$\forall z \in U: \map f z = \map g z$ | {{ProofWanted}}
Category:Complex Analysis
Category:Named Theorems
3mddkpoi5myvk6aq2apc2myvxlfh6vu | Let $U$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Connected Set (Topology)|connected]] [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]] $\C$.
Let $f$ and $g$ be [[Definition:Complex Function|complex functions]] whose [[Definition:Domain of Mapping|domain]] is $U$.... | {{ProofWanted}}
[[Category:Complex Analysis]]
[[Category:Named Theorems]]
3mddkpoi5myvk6aq2apc2myvxlfh6vu | Identity Theorem | https://proofwiki.org/wiki/Identity_Theorem | https://proofwiki.org/wiki/Identity_Theorem | [
"Complex Analysis",
"Named Theorems"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Connected Set (Topology)",
"Definition:Subset",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Function",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Analytic Function",
"Definition:Limit Point/Complex Analysis"
] | [
"Category:Complex Analysis",
"Category:Named Theorems"
] |
proofwiki-11446 | Exponential Sequence is Uniformly Convergent on Compact Sets | Let $\EE = \sequence {E_n}$ denote the sequence of complex functions $E_n: \C \to \C$ defined as:
:$\map {E_n} z = \paren {1 + \dfrac z n}^n$
Let $K$ be a compact subset of $\C$.
Then $\EE$ is uniformly convergent on $K$. | === $\EE$ is Uniformly Bounded on an open space containing $K$ ===
First, from Equivalence of Definitions of Complex Exponential Function we see that $\EE$ is pointwise convergent to $\exp$.
{{refactor|extract this result from that page|level = medium}}
From Combination Theorem for Continuous Complex Functions, $E_n$ ... | Let $\EE = \sequence {E_n}$ denote the [[Definition:Sequence|sequence]] of [[Definition:Complex Function|complex functions]] $E_n: \C \to \C$ defined as:
:$\map {E_n} z = \paren {1 + \dfrac z n}^n$
Let $K$ be a [[Definition:Compact Subset of Complex Plane|compact subset]] of $\C$.
Then $\EE$ is [[Definition:Uniform ... | === $\EE$ is Uniformly Bounded on an open space containing $K$ ===
First, from [[Equivalence of Definitions of Complex Exponential Function]] we see that $\EE$ is [[Definition:Pointwise Convergence|pointwise convergent]] to $\exp$.
{{refactor|extract this result from that page|level = medium}}
From [[Combination Theo... | Exponential Sequence is Uniformly Convergent on Compact Sets | https://proofwiki.org/wiki/Exponential_Sequence_is_Uniformly_Convergent_on_Compact_Sets | https://proofwiki.org/wiki/Exponential_Sequence_is_Uniformly_Convergent_on_Compact_Sets | [
"Exponential Function"
] | [
"Definition:Sequence",
"Definition:Complex Function",
"Definition:Compact Space/Metric Space/Complex",
"Definition:Uniform Convergence/Metric Space"
] | [
"Equivalence of Definitions of Complex Exponential Function",
"Definition:Pointwise Convergence",
"Combination Theorem for Continuous Functions/Complex",
"Definition:Continuous Complex Function",
"Compact Subspace of Metric Space is Bounded",
"Definition:Bounded Metric Space",
"Definition:Real Number",
... |
proofwiki-11447 | Uniformly Convergent Sequence Evaluated on Convergent Sequence | Let $X = \struct {A, d_X}$ and $Y = \struct {B, d_Y}$ be metric spaces.
Let $K$ be a subspace of $X$.
Let $f: X \to Y$ be a mapping.
Let $\FF = \sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to Y$ that converges to $f$ uniformly on $K$.
Let $\sequence {a_n}$ be a convergent sequence in $K$ with limit $... | We want to show that:
:$\map {d_Y} {\map {f_n} {a_n} , \map f a} \to 0$ as $n \to \infty$
Let $\epsilon \in \R_{>0}$ be fixed.
From {{Metric-space-axiom|2}}:
:$\map {d_Y} {\map {f_n} {a_n} , \map f a} \le \map {d_Y} {\map {f_n} {a_n} , \map f {a_n} } + \map {d_Y} {\map f {a_n} , \map f a}$
From the Uniform Limit Theore... | Let $X = \struct {A, d_X}$ and $Y = \struct {B, d_Y}$ be [[Definition:Metric Space|metric spaces]].
Let $K$ be a [[Definition:Metric Subspace|subspace]] of $X$.
Let $f: X \to Y$ be a [[Definition:Mapping|mapping]].
Let $\FF = \sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Continuous Mapping (... | We want to show that:
:$\map {d_Y} {\map {f_n} {a_n} , \map f a} \to 0$ as $n \to \infty$
Let $\epsilon \in \R_{>0}$ be fixed.
From {{Metric-space-axiom|2}}:
:$\map {d_Y} {\map {f_n} {a_n} , \map f a} \le \map {d_Y} {\map {f_n} {a_n} , \map f {a_n} } + \map {d_Y} {\map f {a_n} , \map f a}$
From the [[Uniform Limit... | Uniformly Convergent Sequence Evaluated on Convergent Sequence | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_Evaluated_on_Convergent_Sequence | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_Evaluated_on_Convergent_Sequence | [
"Metric Spaces",
"Uniform Convergence"
] | [
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Mapping",
"Definition:Sequence",
"Definition:Continuous Mapping (Metric Space)",
"Definition: Uniform Convergence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Definition:Sequence",
... | [
"Uniform Limit Theorem",
"Definition:Continuous Mapping (Metric Space)",
"Sequential Continuity is Equivalent to Continuity in Metric Space",
"Definition: Uniform Convergence",
"Category:Metric Spaces",
"Category:Uniform Convergence"
] |
proofwiki-11448 | Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Then
:$\mathscr S$ is meet-continuous {{iff}}
:for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$ | === Sufficient Condition ===
Let $\mathscr S$ be meet-continuous.
Define $\II$, the set of all ideals in $\mathscr S$
Define a mapping $f: \II \to S$ such that
:$\forall I \in \II: \map f I = \sup I$
By Meet-Continuous iff Ideal Supremum is Meet Preserving:
:$f$ preserves meet.
Let $I, J \in \II$.
By definition of mapp... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Then
:$\mathscr S$ is [[Definition:Meet-Continuous Lattice|meet-continuous]] {{iff}}
:for every [[Definition:Ideal in Ordered Set|ideals]] $I, J$ in $\mathscr S$: $\paren {\... | === Sufficient Condition ===
Let $\mathscr S$ be [[Definition:Meet-Continuous Lattice|meet-continuous]].
Define $\II$, the [[Definition:Set of Sets|set]] of all [[Definition:Ideal in Ordered Set|ideals]] in $\mathscr S$
Define a [[Definition:Mapping|mapping]] $f: \II \to S$ such that
:$\forall I \in \II: \map f I = ... | Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_of_Suprema_equals_Supremum_of_Meet_of_Ideals | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_of_Suprema_equals_Supremum_of_Meet_of_Ideals | [
"Meet-Continuous Lattices"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Meet-Continuous Lattice",
"Definition:Ideal in Ordered Set"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set",
"Definition:Mapping",
"Meet-Continuous iff Ideal Supremum is Meet Preserving",
"Definition:Mapping Preserves Infimum/Meet",
"Definition:Mapping Preserves Infimum/Meet",
"Definition:Mapping Preserves Infi... |
proofwiki-11449 | Area between Radii and Whorls of Archimedean Spiral | Let $S$ be the Archimedean spiral defined by the equation:
:$r = a \theta$
Let $\theta = \theta_1$ and $\theta = \theta_2$ be the two rays from the pole at angles $\theta_1$ and $\theta_b$ to the polar axis respectively.
Let $R$ be the figure enclosed by:
:$\theta_1$ and $\theta_2$
:the $n$th turn of $S$ and the $n+1$t... | The straight line boundaries of $R$ are given as $\theta_1$ and $\theta_2$.
The corners of $R$ are located where:
:$\theta = \theta_1 + 2 n \pi$
:$\theta = \theta_2 + 2 n \pi$
:$\theta = \theta_1 + 2 \paren {n + 1} \pi$
:$\theta = \theta_2 + 2 \paren {n + 1} \pi$
500px
{{begin-eqn}}
{{eqn | l = \AA
| r = \int_{\t... | Let $S$ be the [[Definition:Archimedean Spiral|Archimedean spiral]] defined by the equation:
:$r = a \theta$
Let $\theta = \theta_1$ and $\theta = \theta_2$ be the two [[Definition:Ray (Geometry)|rays]] from the [[Definition:Pole (Polar Coordinates)|pole]] at [[Definition:Angle|angles]] $\theta_1$ and $\theta_b$ to t... | The [[Definition:Straight Line|straight line]] boundaries of $R$ are given as $\theta_1$ and $\theta_2$.
The corners of $R$ are located where:
:$\theta = \theta_1 + 2 n \pi$
:$\theta = \theta_2 + 2 n \pi$
:$\theta = \theta_1 + 2 \paren {n + 1} \pi$
:$\theta = \theta_2 + 2 \paren {n + 1} \pi$
[[File:ArchimedeanSpiral... | Area between Radii and Whorls of Archimedean Spiral | https://proofwiki.org/wiki/Area_between_Radii_and_Whorls_of_Archimedean_Spiral | https://proofwiki.org/wiki/Area_between_Radii_and_Whorls_of_Archimedean_Spiral | [
"Archimedean Spiral"
] | [
"Definition:Archimedean Spiral",
"Definition:Line/Infinite Half-Line",
"Definition:Polar Coordinates/Pole",
"Definition:Angle",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Geometric Figure",
"Definition:Area"
] | [
"Definition:Line/Straight Line",
"File:ArchimedeanSpiralAreaBetweenRadii.png",
"Area between Radii and Curve in Polar Coordinates",
"Primitive of Power"
] |
proofwiki-11450 | Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded | Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces.
Let $\left \langle{f_i}\right \rangle_{i \in I}$ be a uniformly convergent sequence of mappings $f_i: X \to Y$.
$\forall i \in I$, let $f_i$ be bounded.
Then $\left \langle{f_i}\right \rangle$ is uniformly bounded. | {{ProofWanted}}
Category:Metric Spaces
7d5j0zwent5gjtl0jwe89zyrpjqlp4t | Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $\left \langle{f_i}\right \rangle_{i \in I}$ be a [[Definition:Uniform Convergence|uniformly convergent]] [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_i: X \to Y$.
$\forall i \in ... | {{ProofWanted}}
[[Category:Metric Spaces]]
7d5j0zwent5gjtl0jwe89zyrpjqlp4t | Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_of_Bounded_Functions_is_Uniformly_Bounded | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_of_Bounded_Functions_is_Uniformly_Bounded | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Uniform Convergence",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Bounded Mapping/Metric Space",
"Definition:Uniformly Bounded"
] | [
"Category:Metric Spaces"
] |
proofwiki-11451 | Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent | Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces.
Let $\sequence {f_n}$ and $\sequence {g_n}$ be sequences of mappings from $X$ to $Y$.
Let $\sequence {f_n}$ and $\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.
$\forall n \in \N$, let $f_n$ and $g_n$ be bounded.
Then the sequ... | {{ProofWanted}}
Category:Metric Spaces
i5xrjd237orij2ge9yvscxqz4r2g3zo | Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be [[Definition:Metric Space|metric spaces]].
Let $\sequence {f_n}$ and $\sequence {g_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Mapping|mappings]] from $X$ to $Y$.
Let $\sequence {f_n}$ and $\sequence {g_n}$ be [[Definition:Uniform Convergence|unifor... | {{ProofWanted}}
[[Category:Metric Spaces]]
i5xrjd237orij2ge9yvscxqz4r2g3zo | Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent | https://proofwiki.org/wiki/Product_of_Uniformly_Convergent_Sequences_of_Bounded_Functions_is_Uniformly_Convergent | https://proofwiki.org/wiki/Product_of_Uniformly_Convergent_Sequences_of_Bounded_Functions_is_Uniformly_Convergent | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Uniform Convergence",
"Definition:Metric Subspace",
"Definition:Bounded Mapping/Metric Space"
] | [
"Category:Metric Spaces"
] |
proofwiki-11452 | Area between Radii and Curve in Polar Coordinates | Let $C$ be a curve expressed in polar coordinates $\polar {r, \theta}$ as:
:$r = \map g \theta$
where $g$ is a real function.
Let $\theta = \theta_a$ and $\theta = \theta_b$ be the two rays from the pole at angles $\theta_a$ and $\theta_b$ to the polar axis respectively.
Then the area $\AA$ between $\theta_a$, $\theta_... | :600px
{{MissingLinks}}
{{tidy|In particular the terms are to be defined.}}
{{improve|We haven't covered NSA on {{ProofWiki}}, and for a result as trivial as this one it seems like overkill.}}
Consider the area of the brown triangle.
This would be:
:$a_\triangle = \dfrac 1 2 r^2 \map \sin {\delta \theta}$
We will be us... | Let $C$ be a [[Definition:Curve|curve]] expressed in [[Definition:Polar Coordinates|polar coordinates]] $\polar {r, \theta}$ as:
:$r = \map g \theta$
where $g$ is a [[Definition:Real Function|real function]].
Let $\theta = \theta_a$ and $\theta = \theta_b$ be the two [[Definition:Ray (Geometry)|rays]] from the [[Def... | :[[File:AreaPolarIntegral.png|600px]]
{{MissingLinks}}
{{tidy|In particular the terms are to be defined.}}
{{improve|We haven't covered NSA on {{ProofWiki}}, and for a result as trivial as this one it seems like overkill.}}
Consider the area of the brown triangle.
This would be:
:$a_\triangle = \dfrac 1 2 r^2 \map \... | Area between Radii and Curve in Polar Coordinates | https://proofwiki.org/wiki/Area_between_Radii_and_Curve_in_Polar_Coordinates | https://proofwiki.org/wiki/Area_between_Radii_and_Curve_in_Polar_Coordinates | [
"Area Formulas"
] | [
"Definition:Line/Curve",
"Definition:Polar Coordinates",
"Definition:Real Function",
"Definition:Line/Infinite Half-Line",
"Definition:Polar Coordinates/Pole",
"Definition:Angle",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Area",
"Definition:Integrable Function"
] | [
"File:AreaPolarIntegral.png",
"Definition:Nonstandard Analysis",
"Definition:Infinitesimal",
"Power Series Expansion for Sine Function",
"Power Series Expansion for Sine Function"
] |
proofwiki-11453 | Exponential Sequence is Eventually Increasing | Let $\sequence {E_n}$ be the sequence of real functions $E_n: \R \to \R$ defined as:
:$\map {E_n} x = \paren {1 + \dfrac x n}^n$
Then, for sufficiently large $n \in \N$, $\sequence {\map {E_n} x}$ is increasing {{WRT}} $n$.
That is:
:$\forall x \in \R: \forall n \in \N: n \ge \ceiling {\size x} \implies \map {E_n} x \l... | Fix $x \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = n
| o = \ge
| r = \ceiling {\size x}
}}
{{eqn | ll= \leadsto
| l = n
| o = >
| r = -x
| c = Real Number is between Ceiling Functions and Negative of Absolute Value
}}
{{eqn | ll= \leadsto
| l = 1
| o = >
| r = \frac {-... | Let $\sequence {E_n}$ be the [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $E_n: \R \to \R$ defined as:
:$\map {E_n} x = \paren {1 + \dfrac x n}^n$
Then, for [[Definition:Sufficiently Large|sufficiently large]] $n \in \N$, $\sequence {\map {E_n} x}$ is [[Definition:Increasing Real Se... | Fix $x \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = n
| o = \ge
| r = \ceiling {\size x}
}}
{{eqn | ll= \leadsto
| l = n
| o = >
| r = -x
| c = [[Real Number is between Ceiling Functions]] and [[Negative of Absolute Value]]
}}
{{eqn | ll= \leadsto
| l = 1
| o = >
| r ... | Exponential Sequence is Eventually Increasing | https://proofwiki.org/wiki/Exponential_Sequence_is_Eventually_Increasing | https://proofwiki.org/wiki/Exponential_Sequence_is_Eventually_Increasing | [
"Exponential Function"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Sufficiently Large",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Ceiling Function"
] | [
"Real Number is between Ceiling Functions",
"Negative of Absolute Value",
"Cauchy's Mean Theorem",
"Power Function is Strictly Increasing over Positive Reals/Natural Exponent",
"Definition:Power (Algebra)/Integer",
"Category:Exponential Function"
] |
proofwiki-11454 | Euler's Number: Limit of Sequence implies Base of Logarithm | Let $e$ be Euler's number defined by:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then $e$ is the unique solution to the equation $\map \ln x = 1$.
That is:
:$\map \ln x = 1 \iff x = e$ | First we prove that $e$ is a solution to $\map \ln x = 1$:
{{begin-eqn}}
{{eqn | l = \map \ln e
| r = \map \ln {\lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n }
| c = {{Defof|Euler's Number as Limit of Sequence}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {\map \ln {1 + \frac 1 n}^n }
| c... | Let $e$ be [[Definition: Euler's Number|Euler's number]] defined by:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then $e$ is the [[Definition:Unique|unique]] solution to the equation $\map \ln x = 1$.
That is:
:$\map \ln x = 1 \iff x = e$ | First we prove that $e$ is a solution to $\map \ln x = 1$:
{{begin-eqn}}
{{eqn | l = \map \ln e
| r = \map \ln {\lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n }
| c = {{Defof|Euler's Number as Limit of Sequence}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {\map \ln {1 + \frac 1 n}^n }
| ... | Euler's Number: Limit of Sequence implies Base of Logarithm | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Base_of_Logarithm | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Base_of_Logarithm | [
"Logarithms",
"Euler's Number"
] | [
"Definition: Euler's Number",
"Definition:Unique"
] | [
"Logarithm of Power",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"L'Hôpital's Rule",
"Logarithm is Strictly Increasing",
"Definition:Strictly Monotone/Mapping",
"Strictly Monotone Mapping with Totally Ordered Domain is Injective",
"Definition:Injection",
"Definition:Unique",... |
proofwiki-11455 | Diophantine Equation y cubed equals x squared plus 2 | The indeterminate Diophantine equation:
:$y^3 = x^2 + 2$
has only one solution in the Natural Numbers:
:$x = 5, y = 3$ | Assume that $x$ is even:
{{begin-eqn}}
{{eqn | l = \paren {2 k}^2 + 2
| r = 4 k^2 + 2
}}
{{eqn | r = 2 \paren {2 k^2 + 1}
| c =
}}
{{end-eqn}}
Therefore, the {{RHS}} is $2 \paren {2 k^2 + 1} \equiv 2 \pmod 4$
If $y$ is odd, then the {{LHS}} will be odd:
{{begin-eqn}}
{{eqn | l = \paren {2 k + 1}^3
| ... | The [[Definition:Indeterminate Equation|indeterminate]] [[Definition:Diophantine Equation|Diophantine equation]]:
:$y^3 = x^2 + 2$
has only one solution in the [[Definition:Natural Numbers|Natural Numbers]]:
:$x = 5, y = 3$ | Assume that $x$ is [[Definition:Even Integer|even]]:
{{begin-eqn}}
{{eqn | l = \paren {2 k}^2 + 2
| r = 4 k^2 + 2
}}
{{eqn | r = 2 \paren {2 k^2 + 1}
| c =
}}
{{end-eqn}}
Therefore, the {{RHS}} is $2 \paren {2 k^2 + 1} \equiv 2 \pmod 4$
If $y$ is [[Definition:Odd Integer|odd]], then the {{LHS}} will be ... | Diophantine Equation y cubed equals x squared plus 2 | https://proofwiki.org/wiki/Diophantine_Equation_y_cubed_equals_x_squared_plus_2 | https://proofwiki.org/wiki/Diophantine_Equation_y_cubed_equals_x_squared_plus_2 | [
"Diophantine Equations"
] | [
"Definition:Indeterminate Equation",
"Definition:Diophantine Equation",
"Definition:Natural Numbers"
] | [
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"D... |
proofwiki-11456 | Continuous Extension from Dense Subset | Let $X$ be a metric space.
Let $D$ be a dense subset of $X$.
Let $f: D \to \R$ be a uniformly continuous mapping.
Then there exists a unique continuous extension of $f$ to $X$. | {{ProofWanted}}
Category:Real Analysis
Category:Metric Spaces
jde7psdzy42exqmb2yjik6axp2bc7ye | Let $X$ be a [[Definition:Metric Space|metric space]].
Let $D$ be a [[Definition:Everywhere Dense|dense]] subset of $X$.
Let $f: D \to \R$ be a [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous mapping]].
Then there exists a unique [[Definition:Continuous Extension|continuous extension]... | {{ProofWanted}}
[[Category:Real Analysis]]
[[Category:Metric Spaces]]
jde7psdzy42exqmb2yjik6axp2bc7ye | Continuous Extension from Dense Subset | https://proofwiki.org/wiki/Continuous_Extension_from_Dense_Subset | https://proofwiki.org/wiki/Continuous_Extension_from_Dense_Subset | [
"Real Analysis",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Everywhere Dense",
"Definition:Uniform Continuity/Metric Space",
"Definition:Continuous Extension"
] | [
"Category:Real Analysis",
"Category:Metric Spaces"
] |
proofwiki-11457 | Sum of two Fourth Powers cannot be Fourth Power | $\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions. | This is a direct consequence of Fermat's Right Triangle Theorem.
{{qed}} | $\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions. | This is a direct consequence of [[Fermat's Right Triangle Theorem]].
{{qed}} | Sum of two Fourth Powers cannot be Fourth Power | https://proofwiki.org/wiki/Sum_of_two_Fourth_Powers_cannot_be_Fourth_Power | https://proofwiki.org/wiki/Sum_of_two_Fourth_Powers_cannot_be_Fourth_Power | [
"Number Theory"
] | [] | [
"Fermat's Right Triangle Theorem"
] |
proofwiki-11458 | Moore-Osgood Theorem | Let $X$ and $Y$ be metric spaces.
Let $S$ be a subspace of $X$.
Let $c$ be a limit point of $S$.
Let $\sequence {f_n}$ be a sequence of mappings $f_n : X \to Y$.
Suppose that:
:$(1): \quad \sequence {f_n}$ is uniformly convergent on $S$
:$(2): \quad \ds \forall n \in \N : \lim_{x \mathop \to c} \map {f_n} x$ exists
The... | {{ProofWanted}}
{{Namedfor|Eliakim Hastings Moore|name2 = William Fogg Osgood|cat = Moore, Eliakim|cat2 = Osgood}}
Category:Real Analysis
Category:Metric Spaces
qgaeu2tjdpwt5olrbi4l67r2ex0ufe6 | Let $X$ and $Y$ be [[Definition: Metric Space|metric spaces]].
Let $S$ be a [[Definition:Metric Subspace|subspace]] of $X$.
Let $c$ be a [[Definition:Limit Point (Metric Space)|limit point]] of $S$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n : X \to Y$.
Suppo... | {{ProofWanted}}
{{Namedfor|Eliakim Hastings Moore|name2 = William Fogg Osgood|cat = Moore, Eliakim|cat2 = Osgood}}
[[Category:Real Analysis]]
[[Category:Metric Spaces]]
qgaeu2tjdpwt5olrbi4l67r2ex0ufe6 | Moore-Osgood Theorem | https://proofwiki.org/wiki/Moore-Osgood_Theorem | https://proofwiki.org/wiki/Moore-Osgood_Theorem | [
"Real Analysis",
"Metric Spaces"
] | [
"Definition: Metric Space",
"Definition:Metric Subspace",
"Definition:Limit Point/Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Uniform Convergence/Metric Space"
] | [
"Category:Real Analysis",
"Category:Metric Spaces"
] |
proofwiki-11459 | Cycloid is Best Shape for Arch | The optimal shape for an arch is a cycloid. | {{ProofWanted|... and of course an elaboration of what "optimal" means would be useful.}} | The optimal shape for an [[Definition:Arch|arch]] is a [[Definition:Cycloid|cycloid]]. | {{ProofWanted|... and of course an elaboration of what "optimal" means would be useful.}} | Cycloid is Best Shape for Arch | https://proofwiki.org/wiki/Cycloid_is_Best_Shape_for_Arch | https://proofwiki.org/wiki/Cycloid_is_Best_Shape_for_Arch | [
"Architecture"
] | [
"Definition:Arch",
"Definition:Cycloid"
] | [] |
proofwiki-11460 | Ore Graph is Connected | Let $G = \struct {V, E}$ be an Ore graph.
Then $G$ is connected. | Let $G$ be an Ore graph of order $n$.
{{AimForCont}} $G$ is not connected.
Then it has at least two components.
Call these components $C_1$ and $C_2$.
Thus, there exist non-adjacent vertices $u$ and $v$ such that $u$ is in $C_1$ and $v$ is in $C_2$.
Let $m_1$ and $m_2$ be the number of vertices in $C_1$ and $C_2$ respe... | Let $G = \struct {V, E}$ be an [[Definition:Ore Graph|Ore graph]].
Then $G$ is [[Definition:Connected Graph|connected]]. | Let $G$ be an [[Definition:Ore Graph|Ore graph]] of [[Definition:Order of Graph|order]] $n$.
{{AimForCont}} $G$ is not [[Definition:Connected Graph|connected]].
Then it has at least two [[Definition:Component of Graph|components]].
Call these [[Definition:Component of Graph|components]] $C_1$ and $C_2$.
Thus, there... | Ore Graph is Connected | https://proofwiki.org/wiki/Ore_Graph_is_Connected | https://proofwiki.org/wiki/Ore_Graph_is_Connected | [
"Ore Graphs"
] | [
"Definition:Ore Graph",
"Definition:Connected (Graph Theory)/Graph"
] | [
"Definition:Ore Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Component of Graph",
"Definition:Component of Graph",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Ore Grap... |
proofwiki-11461 | Power Function to Rational Power permits Unique Continuous Extension | Let $a \in \R_{> 0}$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then there exists a unique continuous extension of $f$ to $\R$. | Consider $I_k := \openint {-k} k$ for arbitrary $k \in \N$.
Let $I_k' = I_k \cap \Q$.
Note that, for all $x, y \in I_k'$:
{{begin-eqn}}
{{eqn | l = \size {a^x - a^y}
| r = \size {a^{x - y + y} - a^y}
}}
{{eqn | r = \size {a^{x - y} a^y - a^y}
| c = Powers of Group Elements
}}
{{eqn | r = \size {a^y} \size {... | Let $a \in \R_{> 0}$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f q = a^q$
where $a^q$ denotes [[Definition:Rational Power|$a$ to the power of $q$]].
Then there exists a unique [[Definition:Continuous Extension|continuous extension]] of $f$ to $\R$. | Consider $I_k := \openint {-k} k$ for arbitrary $k \in \N$.
Let $I_k' = I_k \cap \Q$.
Note that, for all $x, y \in I_k'$:
{{begin-eqn}}
{{eqn | l = \size {a^x - a^y}
| r = \size {a^{x - y + y} - a^y}
}}
{{eqn | r = \size {a^{x - y} a^y - a^y}
| c = [[Powers of Group Elements]]
}}
{{eqn | r = \size {a^y} \... | Power Function to Rational Power permits Unique Continuous Extension | https://proofwiki.org/wiki/Power_Function_to_Rational_Power_permits_Unique_Continuous_Extension | https://proofwiki.org/wiki/Power_Function_to_Rational_Power_permits_Unique_Continuous_Extension | [
"Powers"
] | [
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number",
"Definition:Continuous Extension"
] | [
"Powers of Group Elements",
"Absolute Value Function is Completely Multiplicative",
"Power of Positive Real Number is Positive/Rational Number",
"Power Function is Monotone/Rational Number",
"Power Function tends to One as Power tends to Zero/Rational Number",
"Definition:Uniform Continuity/Real Function"... |
proofwiki-11462 | Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Then:
:$\mathscr S$ is meet-continuous
{{iff}}:
:for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$ | === Sufficient Condition ===
Let $\mathscr S$ be meet-continuous.
By Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals:
:for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$
Let $D_1, D_2$ directed subsets of $S$.
By definition ... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Then:
:$\mathscr S$ is [[Definition:Meet-Continuous Lattice|meet-continuous]]
{{iff}}:
:for every [[Definition:Directed Subset|directed subsets]] $D_1, D_2$ of $S$: $\paren ... | === Sufficient Condition ===
Let $\mathscr S$ be [[Definition:Meet-Continuous Lattice|meet-continuous]].
By [[Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals]]:
:for every [[Definition:Ideal in Ordered Set|ideals]] $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \w... | Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_of_Suprema_equals_Supremum_of_Meet_of_Directed_Subsets | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_of_Suprema_equals_Supremum_of_Meet_of_Directed_Subsets | [
"Meet-Continuous Lattices"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Meet-Continuous Lattice",
"Definition:Directed Subset"
] | [
"Definition:Meet-Continuous Lattice",
"Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals",
"Definition:Ideal in Ordered Set",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Supremum of Lower Closure of Set",
"Definition:Supremum of Set",
"D... |
proofwiki-11463 | Brouwerian Lattice iff Meet-Continuous and Distributive | Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Then
:$\mathscr S$ is a Brouwerian lattice
{{iff}}
:$\mathscr S$ is meet-continuous and distributive. | === Sufficient Condition ===
Let $\mathscr S$ be a Brouwerian lattice.
By Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:
:for every $x \in S$, a mapping $f: S \to S$ if $\forall y \in S: f\left({y}\right) = x \wedge y$, then $f$ is lower adjoint of Galois connection
By Shift M... | Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Then
:$\mathscr S$ is a [[Definition:Brouwerian Lattice|Brouwerian lattice]]
{{iff}}
:$\mathscr S$ is [[Definition:Meet-Continuous Lattice|meet-continuous]] and [[Definition:Distributive Lattice|distribut... | === Sufficient Condition ===
Let $\mathscr S$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]].
By [[Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice]]:
:for every $x \in S$, a [[Definition:Mapping|mapping]] $f: S \to S$ if $\forall y \in S: f\left({y}\right) = x \wedg... | Brouwerian Lattice iff Meet-Continuous and Distributive | https://proofwiki.org/wiki/Brouwerian_Lattice_iff_Meet-Continuous_and_Distributive | https://proofwiki.org/wiki/Brouwerian_Lattice_iff_Meet-Continuous_and_Distributive | [
"Brouwerian Lattices",
"Meet-Continuous Lattices",
"Distributive Lattices",
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Brouwerian Lattice",
"Definition:Meet-Continuous Lattice",
"Definition:Distributive Lattice"
] | [
"Definition:Brouwerian Lattice",
"Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice",
"Definition:Mapping",
"Definition:Galois Connection",
"Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum",
"Definition:Complete Lattice",
"Definition:Up-Comple... |
proofwiki-11464 | Power is Well-Defined/Integer | Let $x$ be a non-zero real number.
Let $k$ be an integer.
Then $x^k$ is well-defined. | Fix $x \in \R \setminus \set 0$. | Let $x$ be a [[Definition:Zero (Number)|non-zero]] [[Definition:Real Number|real number]].
Let $k$ be an [[Definition:Integer|integer]].
Then $x^k$ is [[Definition:Well-Defined Operation|well-defined]]. | Fix $x \in \R \setminus \set 0$. | Power is Well-Defined/Integer | https://proofwiki.org/wiki/Power_is_Well-Defined/Integer | https://proofwiki.org/wiki/Power_is_Well-Defined/Integer | [
"Powers"
] | [
"Definition:Zero (Number)",
"Definition:Real Number",
"Definition:Integer",
"Definition:Well-Defined/Operation"
] | [] |
proofwiki-11465 | Power is Well-Defined/Rational | Let $x \in \R_{> 0}$ be a (strictly) positive real number.
Let $q$ be a rational number.
Then $x^q$ is well-defined. | Let $x \in \R_{>0}$ be fixed.
Let $q \in \Q \setminus \set 0$.
Let $\dfrac r s$ and $\dfrac t u$ be two representations of $q$.
That is, $r, s, t, u$ are non-zero integers.
We now show that:
: $\dfrac r s = \dfrac t u \implies x^{r / s} = x^{t / u}$
So:
{{begin-eqn}}
{{eqn | l = \dfrac r s
| r = \dfrac t u
}}
{{e... | Let $x \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $q$ be a [[Definition:Rational Number|rational number]].
Then $x^q$ is [[Definition:Well-Defined Operation | well-defined]]. | Let $x \in \R_{>0}$ be fixed.
Let $q \in \Q \setminus \set 0$.
Let $\dfrac r s$ and $\dfrac t u$ be two representations of $q$.
That is, $r, s, t, u$ are [[Definition:Zero (Number)|non-zero]] [[Definition:Integer|integers]].
We now show that:
: $\dfrac r s = \dfrac t u \implies x^{r / s} = x^{t / u}$
So:
{{begin... | Power is Well-Defined/Rational | https://proofwiki.org/wiki/Power_is_Well-Defined/Rational | https://proofwiki.org/wiki/Power_is_Well-Defined/Rational | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Rational Number",
"Definition:Well-Defined Operation "
] | [
"Definition:Zero (Number)",
"Definition:Integer",
"Power is Well-Defined/Integer",
"Product of Indices of Real Number/Integers",
"Existence and Uniqueness of Positive Root of Positive Real Number",
"Existence and Uniqueness of Positive Root of Positive Real Number",
"Root is Commutative",
"Category:Po... |
proofwiki-11466 | Real Number between Zero and One is Greater than Power/Natural Number | Let $x \in \R$.
Let $0 < x < 1$.
Let $n$ be a natural number.
Then:
: $0 < x^n \le x$ | For all $n \in \N$, let $\map P n$ be the proposition:
:$0 < x < 1 \implies 0 < x^n \le x$
=== Basis for the Induction ===
$\map P 1$ is true, since $0 < x < 1 \implies 0 < x^1 \le x$ by definition of exponent of $1$.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P... | Let $x \in \R$.
Let $0 < x < 1$.
Let $n$ be a [[Definition:Natural Number|natural number]].
Then:
: $0 < x^n \le x$ | For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$0 < x < 1 \implies 0 < x^n \le x$
=== Basis for the Induction ===
$\map P 1$ is true, since $0 < x < 1 \implies 0 < x^1 \le x$ by definition of exponent of $1$.
This is our [[Definition:Basis for the Induction|basis for the inducti... | Real Number between Zero and One is Greater than Power/Natural Number/Proof 1 | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Power/Natural_Number | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Power/Natural_Number/Proof_1 | [
"Powers",
"Real Numbers",
"Inequalities",
"Real Number between Zero and One is Greater than Power"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Real Number between Zero and One is Greater than Power/Natural Number/Proof 1",
"Real Number Ordering is Compatible with Multiplication",
"Multiple of Positive Real Number wit... |
proofwiki-11467 | Power Function is Strictly Increasing over Positive Reals/Natural Exponent | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $f: \R_{>0} \to \R$ be the real function defined as:
:$\map f x = x^n$
where $x^n$ denotes $x$ to the power of $n$.
Then $f$ is strictly increasing. | Proof by induction on $n$:
Let $x, y \in \R_{>0}$ be strictly positive real numbers.
For all $n \in \Z_{>0}$, let $P \left({n}\right)$ be the proposition:
:$x < y \implies x^n < y^n$ | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $f: \R_{>0} \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = x^n$
where $x^n$ denotes [[Definition:Integer Power|$x$ to the power of $n$]].
Then $f$ is [[Definition:Strictly Increasing R... | Proof by [[Principle of Mathematical Induction|induction]] on $n$:
Let $x, y \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
For all $n \in \Z_{>0}$, let $P \left({n}\right)$ be the [[Definition:Proposition|proposition]]:
:$x < y \implies x^n < y^n$ | Power Function is Strictly Increasing over Positive Reals/Natural Exponent | https://proofwiki.org/wiki/Power_Function_is_Strictly_Increasing_over_Positive_Reals/Natural_Exponent | https://proofwiki.org/wiki/Power_Function_is_Strictly_Increasing_over_Positive_Reals/Natural_Exponent | [
"Examples of Strictly Increasing Real Functions",
"Powers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Real Function",
"Definition:Power (Algebra)/Integer",
"Definition:Strictly Increasing/Real Function"
] | [
"Principle of Mathematical Induction",
"Definition:Strictly Positive/Real Number",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11468 | Root is Strictly Increasing | Let $x \in \R_{> 0}$.
Let $n \in \N$.
Let $f: \R_{> 0} \to \R$ be the real function defined as:
:$\map f x = \sqrt [n] x$
where $\sqrt [n] x$ denotes the $n$th root of $x$.
Then $f$ is strictly increasing. | Let $x, y \in \R$ such that $0 < x < y$.
{{AimForCont}} that:
:$\sqrt [n] x \ge \sqrt [n] y$
We have:
{{begin-eqn}}
{{eqn | l = \sqrt [n] x
| o = \ge
| r = \sqrt [n] y
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\sqrt [n] x}^n
| o = \ge
| r = \paren {\sqrt [n] y}^n
| c = Power... | Let $x \in \R_{> 0}$.
Let $n \in \N$.
Let $f: \R_{> 0} \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = \sqrt [n] x$
where $\sqrt [n] x$ denotes the [[Definition:Root of Number|$n$th root]] of $x$.
Then $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]]. | Let $x, y \in \R$ such that $0 < x < y$.
{{AimForCont}} that:
:$\sqrt [n] x \ge \sqrt [n] y$
We have:
{{begin-eqn}}
{{eqn | l = \sqrt [n] x
| o = \ge
| r = \sqrt [n] y
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\sqrt [n] x}^n
| o = \ge
| r = \paren {\sqrt [n] y}^n
| c = [[P... | Root is Strictly Increasing | https://proofwiki.org/wiki/Root_is_Strictly_Increasing | https://proofwiki.org/wiki/Root_is_Strictly_Increasing | [
"Examples of Strictly Increasing Real Functions",
"Roots of Numbers"
] | [
"Definition:Real Function",
"Definition:Root of Number",
"Definition:Strictly Increasing/Real Function"
] | [
"Power Function is Strictly Increasing over Positive Reals/Natural Exponent",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Strictly Increasing/Real Function",
"Category:Examples of Strictly Increasing Real Functions",
"Category:Roots of Numbers"
] |
proofwiki-11469 | Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum | Let $\struct {S, \preceq}$ be a complete lattice.
Then:
:$\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is lower adjoint of a Galois connection
{{iff}}:
:$\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \set {x \wedge y: y \in X}$ | === Sufficient Condition ===
Assume that
:$\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is lower adjoint of a Galois connection
Let $x \in S, X \subseteq S$.
Define a mapping $f: S \to S$:
:$\forall y \in S: \map f y := x \wedge y$
By assumption:
:$f$ is lower adjoint of a Ga... | Let $\struct {S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Then:
:$\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is [[Definition:Galois Connection|lower adjoint of a Galois connection]]
{{iff}}:
:$\forall x \in S, X \subseteq S: x \wedge \sup X = \sup ... | === Sufficient Condition ===
Assume that
:$\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is [[Definition:Galois Connection|lower adjoint of a Galois connection]]
Let $x \in S, X \subseteq S$.
Define a [[Definition:Mapping|mapping]] $f: S \to S$:
:$\forall y \in S: \map f y ... | Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum | https://proofwiki.org/wiki/Shift_Mapping_is_Lower_Adjoint_iff_Meet_is_Distributive_with_Supremum | https://proofwiki.org/wiki/Shift_Mapping_is_Lower_Adjoint_iff_Meet_is_Distributive_with_Supremum | [
"Galois Connections",
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Galois Connection"
] | [
"Definition:Galois Connection",
"Definition:Mapping",
"Definition:Galois Connection",
"Lower Adjoint Preserves All Suprema",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Complete Lattice",
"Defin... |
proofwiki-11470 | Greatest Common Divisor is Associative | Let $a, b, c \in \Z$.
Then:
:$\gcd \set {a, \gcd \set {b, c} } = \gcd \set {\gcd \set {a, b}, c}$
where $\gcd$ denotes the greatest common divisor. | Follows directly from GCD from Prime Decomposition and Min Operation is Associative. | Let $a, b, c \in \Z$.
Then:
:$\gcd \set {a, \gcd \set {b, c} } = \gcd \set {\gcd \set {a, b}, c}$
where $\gcd$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]. | Follows directly from [[GCD from Prime Decomposition]] and [[Min Operation is Associative]]. | Greatest Common Divisor is Associative | https://proofwiki.org/wiki/Greatest_Common_Divisor_is_Associative | https://proofwiki.org/wiki/Greatest_Common_Divisor_is_Associative | [
"Greatest Common Divisor",
"Examples of Associative Operations"
] | [
"Definition:Greatest Common Divisor/Integers"
] | [
"GCD from Prime Decomposition",
"Min Operation is Associative"
] |
proofwiki-11471 | Lowest Common Multiple is Associative | Let $a, b, c \in \Z$.
Then:
:$\lcm \set {a, \lcm \set {b, c} } = \lcm \set {\lcm \set {a, b}, c}$
where $\lcm$ denotes the lowest common multiple. | Follows directly from LCM from Prime Decomposition and Max Operation is Associative. | Let $a, b, c \in \Z$.
Then:
:$\lcm \set {a, \lcm \set {b, c} } = \lcm \set {\lcm \set {a, b}, c}$
where $\lcm$ denotes the [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]. | Follows directly from [[LCM from Prime Decomposition]] and [[Max Operation is Associative]]. | Lowest Common Multiple is Associative | https://proofwiki.org/wiki/Lowest_Common_Multiple_is_Associative | https://proofwiki.org/wiki/Lowest_Common_Multiple_is_Associative | [
"Lowest Common Multiple",
"Examples of Associative Operations"
] | [
"Definition:Lowest Common Multiple/Integers"
] | [
"LCM from Prime Decomposition",
"Max Operation is Associative"
] |
proofwiki-11472 | LCM equals Product iff Coprime | Let $a, b \in \Z_{>0}$ be strictly positive integers.
Then:
:$\lcm \set {a, b} = a b$
{{iff}}:
:$a$ and $b$ are coprime
where $\lcm$ denotes the lowest common multiple. | === Necessary Condition ===
Let $a$ and $b$ be coprime.
Then:
{{begin-eqn}}
{{eqn | l = \lcm \set {a, b}
| r = \frac {a b} {\gcd \set {a, b} }
| c = Product of GCD and LCM
}}
{{eqn | r = \frac {a b} 1
| c = {{Defof|Coprime Integers}}
}}
{{eqn | r = a b
}}
{{end-eqn}}
{{qed|lemma}} | Let $a, b \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Then:
:$\lcm \set {a, b} = a b$
{{iff}}:
:$a$ and $b$ are [[Definition:Coprime Integers|coprime]]
where $\lcm$ denotes the [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]. | === Necessary Condition ===
Let $a$ and $b$ be [[Definition:Coprime Integers|coprime]].
Then:
{{begin-eqn}}
{{eqn | l = \lcm \set {a, b}
| r = \frac {a b} {\gcd \set {a, b} }
| c = [[Product of GCD and LCM]]
}}
{{eqn | r = \frac {a b} 1
| c = {{Defof|Coprime Integers}}
}}
{{eqn | r = a b
}}
{{end-e... | LCM equals Product iff Coprime | https://proofwiki.org/wiki/LCM_equals_Product_iff_Coprime | https://proofwiki.org/wiki/LCM_equals_Product_iff_Coprime | [
"Lowest Common Multiple",
"Coprime Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Lowest Common Multiple/Integers"
] | [
"Definition:Coprime/Integers",
"Product of GCD and LCM",
"Product of GCD and LCM"
] |
proofwiki-11473 | GCD with One Fixed Argument is Multiplicative Function | Let:
:$a, b, c \in \Z: b \perp c$
where $b \perp c$ denotes that $b$ is coprime to $c$.
Then:
:$\gcd \set {a, b} \gcd \set {a, c} = \gcd \set {a, b c}$
That is, GCD is multiplicative. | {{begin-eqn}}
{{eqn | l = \gcd \set {a, b c}
| r = \gcd \set {a, \lcm \set {b, c} }
| c = LCM equals Product iff Coprime
}}
{{eqn | r = \lcm \set {\gcd \set {a, b}, \gcd \set {a, c} }
| c = GCD and LCM Distribute Over Each Other
}}
{{eqn | r = \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {\gc... | Let:
:$a, b, c \in \Z: b \perp c$
where $b \perp c$ denotes that $b$ is [[Definition:Coprime Integers|coprime]] to $c$.
Then:
:$\gcd \set {a, b} \gcd \set {a, c} = \gcd \set {a, b c}$
That is, [[Definition:Greatest Common Divisor of Integers|GCD]] is [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | {{begin-eqn}}
{{eqn | l = \gcd \set {a, b c}
| r = \gcd \set {a, \lcm \set {b, c} }
| c = [[LCM equals Product iff Coprime]]
}}
{{eqn | r = \lcm \set {\gcd \set {a, b}, \gcd \set {a, c} }
| c = [[GCD and LCM Distribute Over Each Other]]
}}
{{eqn | r = \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \... | GCD with One Fixed Argument is Multiplicative Function | https://proofwiki.org/wiki/GCD_with_One_Fixed_Argument_is_Multiplicative_Function | https://proofwiki.org/wiki/GCD_with_One_Fixed_Argument_is_Multiplicative_Function | [
"Greatest Common Divisor",
"Examples of Multiplicative Functions"
] | [
"Definition:Coprime/Integers",
"Definition:Greatest Common Divisor/Integers",
"Definition:Multiplicative Arithmetic Function"
] | [
"LCM equals Product iff Coprime",
"GCD and LCM Distribute Over Each Other",
"Product of GCD and LCM",
"Greatest Common Divisor is Associative",
"Greatest Common Divisor is Associative",
"Category:Greatest Common Divisor",
"Category:Examples of Multiplicative Functions"
] |
proofwiki-11474 | Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer | Let $a \in \R$ be a real number such that $0 < a < 1$.
Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as:
:$\map f n = a^n$
where $a^n$ denotes $a$ to the power of $n$.
Then $f$ is strictly decreasing. | Proof by induction on $n$:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$a^{n + 1} < a^n$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = a^1
| r = a
| c = {{Defof|Integer Power}}
}}
{{eqn | o = <
| r = 1
| c =
}}
{{eqn | r = a^0
| c = {{Defof|Integer Power}}
}}
{{en... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $0 < a < 1$.
Let $f: \Z_{\ge 0} \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f n = a^n$
where $a^n$ denotes [[Definition:Integer Power|$a$ to the power of $n$]].
Then $f$ is [[Definition:Strictly Decrea... | Proof by [[Principle of Mathematical Induction|induction]] on $n$:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$a^{n + 1} < a^n$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = a^1
| r = a
| c = {{Defof|Integer Power}}
}}
{{eqn | o = <
| r = 1
... | Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Positive_Integer | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Positive_Integer | [
"Power Function on Base between Zero and One is Strictly Decreasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Integer",
"Definition:Strictly Decreasing/Real Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11475 | Power Function on Base between Zero and One is Strictly Decreasing/Rational Number | Let $a \in \R$ be a real number such that $0 < a < 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then $f$ is strictly decreasing. | Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z, s, u \in \Z_{>0}$.
Let $\dfrac r s < \dfrac t u$.
Then:
{{begin-eqn}}
{{eqn | l = r u
| o = <
| r = t s
| c = Real Number Ordering is Compatible with Multiplication
}}
{{eqn | ll= \leadsto
| l = a^{r u}
| o = >
| r = a^{t s}
... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $0 < a < 1$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function |real-valued function]] defined as:
:$\map f q = a^q$
where $a^q$ denotes [[Definition:Rational Power|$a$ to the power of $q$]].
Then $f$ is [[Definition:Strictly Decreasing R... | Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z, s, u \in \Z_{>0}$.
Let $\dfrac r s < \dfrac t u$.
Then:
{{begin-eqn}}
{{eqn | l = r u
| o = <
| r = t s
| c = [[Real Number Ordering is Compatible with Multiplication]]
}}
{{eqn | ll= \leadsto
| l = a^{r u}
| o = >
| r = a^... | Power Function on Base between Zero and One is Strictly Decreasing/Rational Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Rational_Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Rational_Number | [
"Power Function on Base between Zero and One is Strictly Decreasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function ",
"Definition:Power (Algebra)/Rational Number",
"Definition:Strictly Decreasing/Real Function"
] | [
"Real Number Ordering is Compatible with Multiplication",
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Product of Indices of Real Number/Integers",
"Root is Strictly Increasing",
"Root is Strictly Increasing",
"Root is Commutative",
"Category:Power Function on Base betw... |
proofwiki-11476 | Power Function on Base Greater than One is Strictly Increasing/Positive Integer | Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as:
:$\map f n = a^n$
where $a^n$ denotes $a$ to the power of $n$.
Then $f$ is strictly increasing. | Fix $n \in \Z_{\ge 0}$.
From Ordering of Reciprocals:
:$0 < \dfrac 1 a < 1$
From Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer:
:$\paren {\dfrac 1 a}^{n + 1} < \paren {\dfrac 1 a}^n$
From Real Number to Negative Power: Positive Integer:
:$\dfrac 1 {a^{n + 1} } < \dfrac 1 {a^n}$
Fr... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 1$.
Let $f: \Z_{\ge 0} \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f n = a^n$
where $a^n$ denotes [[Definition:Integer Power|$a$ to the power of $n$]].
Then $f$ is [[Definition:Strictly Increasing... | Fix $n \in \Z_{\ge 0}$.
From [[Ordering of Reciprocals]]:
:$0 < \dfrac 1 a < 1$
From [[Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer|Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer]]:
:$\paren {\dfrac 1 a}^{n + 1} < \paren {\dfrac 1 a}^n$
Fr... | Power Function on Base Greater than One is Strictly Increasing/Positive Integer | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Positive_Integer | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Positive_Integer | [
"Power Function on Base Greater than One is Strictly Increasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Integer",
"Definition:Strictly Increasing/Real Function"
] | [
"Ordering of Reciprocals",
"Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer",
"Real Number to Negative Power/Positive Integer",
"Ordering of Reciprocals",
"Category:Power Function on Base Greater than One is Strictly Increasing"
] |
proofwiki-11477 | Root of Reciprocal is Reciprocal of Root | Let $x \in \R_{\ge 0}$.
Let $n \in \N$.
Let $\sqrt [n] x$ denote the $n$th root of $x$.
Then:
:$\sqrt [n] {\dfrac 1 x} = \dfrac 1 {\sqrt [n] x}$ | Let $y = \sqrt [n] {\dfrac 1 x}$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt [n] {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 x
| r = y^n
| c = {{Defof|Root of Number|$n$th root}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \dfrac 1 {y^n}
| c = Re... | Let $x \in \R_{\ge 0}$.
Let $n \in \N$.
Let $\sqrt [n] x$ denote the [[Definition:Root of Number|$n$th root]] of $x$.
Then:
:$\sqrt [n] {\dfrac 1 x} = \dfrac 1 {\sqrt [n] x}$ | Let $y = \sqrt [n] {\dfrac 1 x}$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt [n] {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 x
| r = y^n
| c = {{Defof|Root of Number|$n$th root}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \dfrac 1 {y^n}
| c = ... | Root of Reciprocal is Reciprocal of Root | https://proofwiki.org/wiki/Root_of_Reciprocal_is_Reciprocal_of_Root | https://proofwiki.org/wiki/Root_of_Reciprocal_is_Reciprocal_of_Root | [
"Roots of Numbers"
] | [
"Definition:Root of Number"
] | [
"Reciprocal of Real Number is Unique",
"Powers of Group Elements/Negative Index",
"Reciprocal of Real Number is Unique",
"Category:Roots of Numbers"
] |
proofwiki-11478 | Power Function on Base Greater than One is Strictly Increasing/Rational Number | Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then $f$ is strictly increasing. | Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z$ are integers and $s, u \in \Z_{>0}$ are strictly positive integers.
Let $\dfrac r s < \dfrac t u$.
From Ordering of Reciprocals:
:$0 < \dfrac 1 a < 1$
So:
{{begin-eqn}}
{{eqn | l = \paren {\frac 1 a}^{t / u}
| o = <
| r = \paren {\frac 1 a}^{r / s}
... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 1$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f q = a^q$
where $a^q$ denotes [[Definition:Rational Power|$a$ to the power of $q$]].
Then $f$ is [[Definition:Strictly Increasing Real F... | Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z$ are [[Definition:Integer|integers]] and $s, u \in \Z_{>0}$ are [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $\dfrac r s < \dfrac t u$.
From [[Ordering of Reciprocals]]:
:$0 < \dfrac 1 a < 1$
So:
{{begin-eqn}}
{{eqn | l = \paren {\... | Power Function on Base Greater than One is Strictly Increasing/Rational Number | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Rational_Number | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Rational_Number | [
"Power Function on Base Greater than One is Strictly Increasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number",
"Definition:Strictly Increasing/Real Function"
] | [
"Definition:Integer",
"Definition:Strictly Positive/Integer",
"Ordering of Reciprocals",
"Power Function on Base between Zero and One is Strictly Decreasing/Rational Number",
"Real Number to Negative Power/Integer",
"Root of Reciprocal is Reciprocal of Root",
"Ordering of Reciprocals",
"Category:Power... |
proofwiki-11479 | Existence of Square Roots of Positive Real Number | Let $r \in \R_{\ge 0}$ be a positive real number.
Then:
:$\exists y_1 \in \R_{\ge 0}: {y_1}^2 = r$
:$\exists y_2 \in \R_{\le 0}: {y_2}^2 = r$ | Let $S = \set {x \in \R: x^2 < r}$.
As $0 \in S$, it follows that $S$ is non-empty.
To show that $S$ is bounded above, we note that $r + 1$ is an upper bound:
:$y > r + 1 \implies y^2 > r^2 + 2 r + 1 > r$
and so $y \notin S$.
Thus $x \in S \implies x < r + 1$.
By the Completeness Axiom, $S$ has a supremum, say:
:$u = \... | Let $r \in \R_{\ge 0}$ be a [[Definition:Positive Real Number|positive real number]].
Then:
:$\exists y_1 \in \R_{\ge 0}: {y_1}^2 = r$
:$\exists y_2 \in \R_{\le 0}: {y_2}^2 = r$ | Let $S = \set {x \in \R: x^2 < r}$.
As $0 \in S$, it follows that $S$ is [[Definition:Non-Empty Set|non-empty]].
To show that $S$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]], we note that $r + 1$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]]:
:$y > r + 1 \implies y^2 ... | Existence of Square Roots of Positive Real Number | https://proofwiki.org/wiki/Existence_of_Square_Roots_of_Positive_Real_Number | https://proofwiki.org/wiki/Existence_of_Square_Roots_of_Positive_Real_Number | [
"Roots of Numbers",
"Square Roots",
"Existence of Square Roots of Positive Real Number"
] | [
"Definition:Positive/Real Number"
] | [
"Definition:Non-Empty Set",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Contradiction",
"Definition:Contradiction",
"Definition:Upper Bound of Set/Real Numbers",
"Proof by Cont... |
proofwiki-11480 | Inequality iff Difference is Positive | Let $x, y \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$x < y$}}
{{item|(2):|$y - x > 0$}}
{{end-itemize}} | {{begin-eqn}}
{{eqn | l = x < y
| o = \leadstoandfrom
| r = y > x
| c = {{Defof|Dual Ordering}}
}}
{{eqn | o = \leadstoandfrom
| r = y + \paren {-x} > x + \paren {-x}
| c = Real Number Ordering is Compatible with Addition
}}
{{eqn | o = \leadstoandfrom
| r = y + \paren {-x} > 0
... | Let $x, y \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$x < y$}}
{{item|(2):|$y - x > 0$}}
{{end-itemize}} | {{begin-eqn}}
{{eqn | l = x < y
| o = \leadstoandfrom
| r = y > x
| c = {{Defof|Dual Ordering}}
}}
{{eqn | o = \leadstoandfrom
| r = y + \paren {-x} > x + \paren {-x}
| c = [[Real Number Ordering is Compatible with Addition]]
}}
{{eqn | o = \leadstoandfrom
| r = y + \paren {-x} > 0
... | Inequality iff Difference is Positive | https://proofwiki.org/wiki/Inequality_iff_Difference_is_Positive | https://proofwiki.org/wiki/Inequality_iff_Difference_is_Positive | [
"Inequalities",
"Real Numbers"
] | [] | [
"Real Number Ordering is Compatible with Addition"
] |
proofwiki-11481 | Power Function on Base between Zero and One is Strictly Decreasing/Integer | Let $a \in \R$ be a real number such that $0 < a < 1$.
Let $f: \Z \to \R$ be the real-valued function defined as:
:$\map f k = a^k$
where $a^k$ denotes $a$ to the power of $k$.
Then $f$ is strictly decreasing. | Let $0 < a < 1$.
By Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer, the theorem is already proven for positive integers.
It remains to be proven over the negative integers.
Let $i, j$ be integers such that $i < j < 0$.
From Order of Real Numbers is Dual of Order of their Negatives:... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $0 < a < 1$.
Let $f: \Z \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f k = a^k$
where $a^k$ denotes [[Definition:Integer Power|$a$ to the power of $k$]].
Then $f$ is [[Definition:Strictly Decreasing Rea... | Let $0 < a < 1$.
By [[Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer|Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer]], the theorem is already proven for [[Definition:Positive Integer|positive integers]].
It remains to be proven over the [[Defi... | Power Function on Base between Zero and One is Strictly Decreasing/Integer | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Integer | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Integer | [
"Power Function on Base between Zero and One is Strictly Decreasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Integer",
"Definition:Strictly Decreasing/Real Function"
] | [
"Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer",
"Definition:Positive/Integer",
"Definition:Negative/Integer",
"Definition:Integer",
"Order of Real Numbers is Dual of Order of their Negatives",
"Power Function on Base between Zero and One is Strictly Decreasing/Positi... |
proofwiki-11482 | Meet-Continuous iff Meet Preserves Directed Suprema | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Let $\struct {S \times S, \precsim}$ be the simple order product of $\mathscr S$ and $\mathscr S$.
Let $f: S \times S \to S$ be a mapping such that
:$\forall x, y \in S: \map f {x, y} = x \wedge y$
Then:
:$\mathscr S$ is meet-continuous
{{... | === Sufficient Condition ===
Assume that:
:$\mathscr S$ is meet-continuous.
We will prove that:
:for every element $x$ of $S$, a directed subset $D$ of $S$ if $x \preceq \sup D$, then $x = \sup \set {x \wedge d: d \in D}$
Let $x \in S$, $D$ be a directed subset of $S$ such that:
:$x \preceq \sup D$
Thus
{{begin-eqn}}
{... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Let $\struct {S \times S, \precsim}$ be the [[Definition:Simple Order Product|simple order product]] of $\mathscr S$ and $\mathscr S$.
Let $f: S \times S \to S$ be a [[Defi... | === Sufficient Condition ===
Assume that:
:$\mathscr S$ is [[Definition:Meet-Continuous Lattice|meet-continuous]].
We will prove that:
:for every [[Definition:Element|element]] $x$ of $S$, a [[Definition:Directed Subset|directed subset]] $D$ of $S$ if $x \preceq \sup D$, then $x = \sup \set {x \wedge d: d \in D}$
Le... | Meet-Continuous iff Meet Preserves Directed Suprema | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Meet-Continuous_iff_Meet_Preserves_Directed_Suprema | [
"Meet-Continuous Lattices"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Simple Order Product",
"Definition:Mapping",
"Definition:Meet-Continuous Lattice",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Element",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Preceding iff Meet equals Less Operand",
"Definition:Reflexivity",
"Definition:Element",
"Definition:Directed Subset",
"Meet Preserves Directed Suprema",
"Definition:Mapping Pre... |
proofwiki-11483 | Power Function on Base Greater than One is Strictly Increasing/Integer | Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Z \to \R$ be the real-valued function defined as:
:$\map f k = a^k$
where $a^k$ denotes $a$ to the power of $k$.
Then $f$ is strictly decreasing. | Let $a > 1$.
By Power Function on Base Greater than One is Strictly Increasing: Positive Integer, the theorem is already proven for positive integers.
It remains to be proven over the strictly negative integers.
Let $i, j$ be integers such that $i < j < 0$.
From Order of Real Numbers is Dual of Order of their Negatives... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 1$.
Let $f: \Z \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f k = a^k$
where $a^k$ denotes [[Definition:Integer Power|$a$ to the power of $k$]].
Then $f$ is [[Definition:Strictly Decreasing Real Fu... | Let $a > 1$.
By [[Power Function on Base Greater than One is Strictly Increasing/Positive Integer|Power Function on Base Greater than One is Strictly Increasing: Positive Integer]], the theorem is already proven for [[Definition:Positive Integer|positive integers]].
It remains to be proven over the [[Definition:Stric... | Power Function on Base Greater than One is Strictly Increasing/Integer | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Integer | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Integer | [
"Power Function on Base Greater than One is Strictly Increasing"
] | [
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Integer",
"Definition:Strictly Decreasing/Real Function"
] | [
"Power Function on Base Greater than One is Strictly Increasing/Positive Integer",
"Definition:Positive/Integer",
"Definition:Strictly Negative/Integer",
"Definition:Integer",
"Order of Real Numbers is Dual of Order of their Negatives",
"Power Function on Base Greater than One is Strictly Increasing/Posit... |
proofwiki-11484 | Real Star-Algebra is Commutative | Let $A = \struct {A_F, \oplus}$ be a real $*$-algebra whose conjugation is denoted as $*$.
Then:
:$\forall a, b \in A, a \oplus b = b \oplus a$
That is, real $*$-algebra is commutative. | {{begin-eqn}}
{{eqn | l = a \oplus b
| r = \paren {a \oplus b}^*
| c = {{Defof|Real Star-Algebra|Real $*$-Algebra}}
}}
{{eqn | r = b^* \oplus a^*
| c = {{Defof|Conjugation on Algebra}}
}}
{{eqn | r = b \oplus a
| c = {{Defof|Real Star-Algebra|Real $*$-Algebra}}
}}
{{end-eqn}}
{{qed}}
Category:Re... | Let $A = \struct {A_F, \oplus}$ be a [[Definition:Real Star-Algebra|real $*$-algebra]] whose [[Definition:Conjugation on Algebra|conjugation]] is denoted as $*$.
Then:
:$\forall a, b \in A, a \oplus b = b \oplus a$
That is, [[Definition:Real Star-Algebra|real $*$-algebra]] is [[Definition:Commutative Algebra|commuta... | {{begin-eqn}}
{{eqn | l = a \oplus b
| r = \paren {a \oplus b}^*
| c = {{Defof|Real Star-Algebra|Real $*$-Algebra}}
}}
{{eqn | r = b^* \oplus a^*
| c = {{Defof|Conjugation on Algebra}}
}}
{{eqn | r = b \oplus a
| c = {{Defof|Real Star-Algebra|Real $*$-Algebra}}
}}
{{end-eqn}}
{{qed}}
[[Category... | Real Star-Algebra is Commutative | https://proofwiki.org/wiki/Real_Star-Algebra_is_Commutative | https://proofwiki.org/wiki/Real_Star-Algebra_is_Commutative | [
"Real Star-Algebras",
"Commutative Algebras"
] | [
"Definition:Real Star-Algebra",
"Definition:Conjugation on Algebra",
"Definition:Real Star-Algebra",
"Definition:Commutative Algebra"
] | [
"Category:Real Star-Algebras",
"Category:Commutative Algebras"
] |
proofwiki-11485 | Meet-Continuous implies Shift Mapping Preserves Directed Suprema | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a meet-continuous lattice.
Let $x \in S$.
Let $f: S \to S$ be a mapping such that:
:$\forall y \in S: \map f y = x \wedge y$
Then:
:$f$ preserves directed suprema. | Let $D$ be a directed subset of $S$ such that
:$D$ admits a supremum.
By Singleton is Directed and Filtered Subset:
:$\set x$ is directed.
By Up-Complete Product/Lemma 1:
:$\set x \times D$ is directed in the simple order product $\struct {S \times S, \precsim}$ of $\mathscr S$ and $\mathscr S$.
Define a mapping $g: S ... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Meet-Continuous Lattice|meet-continuous lattice]].
Let $x \in S$.
Let $f: S \to S$ be a [[Definition:Mapping|mapping]] such that:
:$\forall y \in S: \map f y = x \wedge y$
Then:
:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves di... | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$D$ admits a [[Definition:Supremum of Set|supremum]].
By [[Singleton is Directed and Filtered Subset]]:
:$\set x$ is [[Definition:Directed Subset|directed]].
By [[Up-Complete Product/Lemma 1]]:
:$\set x \times D$ is [[Definition:Directed Su... | Meet-Continuous implies Shift Mapping Preserves Directed Suprema | https://proofwiki.org/wiki/Meet-Continuous_implies_Shift_Mapping_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Meet-Continuous_implies_Shift_Mapping_Preserves_Directed_Suprema | [
"Meet-Continuous Lattices",
"Suprema"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Singleton is Directed and Filtered Subset",
"Definition:Directed Subset",
"Up-Complete Product/Lemma 1",
"Definition:Directed Subset",
"Definition:Simple Order Product",
"Definition:Mapping",
"Meet-Continuous iff Meet Preserves Directed Su... |
proofwiki-11486 | Power of Strictly Positive Real Number is Strictly Positive/Positive Integer | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$x^n > 0$
where $x^n$ denotes the $n$th power of $x$. | Proof by Mathematical Induction:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\forall x \in \R_{>0}: x^n > 0$
$\map P 0$ is true, as this just says:
{{begin-eqn}}
{{eqn | l = x^0
| r = 1
| c = {{Defof|Integer Power}}
}}
{{eqn | o = >
| r = 0
}}
{{end-eqn}} | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$x^n > 0$
where $x^n$ denotes the [[Definition:Integer Power|$n$th power of $x$]]. | Proof by [[Proof by Mathematical Induction|Mathematical Induction]]:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall x \in \R_{>0}: x^n > 0$
$\map P 0$ is true, as this just says:
{{begin-eqn}}
{{eqn | l = x^0
| r = 1
| c = {{Defof|Integer Power}}
}}
{{... | Power of Strictly Positive Real Number is Strictly Positive/Positive Integer | https://proofwiki.org/wiki/Power_of_Strictly_Positive_Real_Number_is_Strictly_Positive/Positive_Integer | https://proofwiki.org/wiki/Power_of_Strictly_Positive_Real_Number_is_Strictly_Positive/Positive_Integer | [
"Power of Positive Real Number is Positive"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Positive/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-11487 | Power of Positive Real Number is Positive/Integer | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $n \in \Z$ be an integer.
Then:
:$x^n > 0$
where $x^n$ denotes the $n$th power of $x$. | By Power of Positive Real Number is Positive: Natural Number, the theorem is already proven for non-negative integers.
Suppose $n \in \Z_{< 0}$.
When $n < 0$, by Real Number Ordering is Compatible with Multiplication: Negative Factor:
:$-n > 0$
Then, by Power of Positive Real Number is Positive: Natural Number:
:$x^{-n... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$x^n > 0$
where $x^n$ denotes the [[Definition:Integer Power|$n$th power of $x$]]. | By [[Power of Positive Real Number is Positive/Natural Number|Power of Positive Real Number is Positive: Natural Number]], the theorem is already proven for [[Definition:Non-Negative Integer|non-negative integers]].
Suppose $n \in \Z_{< 0}$.
When $n < 0$, by [[Real Number Ordering is Compatible with Multiplication/Ne... | Power of Positive Real Number is Positive/Integer | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Integer | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Integer | [
"Power of Positive Real Number is Positive"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Power of Strictly Positive Real Number is Strictly Positive/Positive Integer",
"Definition:Positive/Integer",
"Real Number Ordering is Compatible with Multiplication/Negative Factor",
"Power of Strictly Positive Real Number is Strictly Positive/Positive Integer",
"Reciprocal of Strictly Positive Real Numbe... |
proofwiki-11488 | Power of Positive Real Number is Positive/Rational Number | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $q \in \Q$ be a rational number.
Then:
:$x^q > 0$
where $x^q$ denotes the $x$ to the power of $q$. | Let $q = \dfrac r s$, where $r \in \Z$, $s \in \Z \setminus \set 0$.
Then:
{{begin-eqn}}
{{eqn | l = x > 0
| o = \leadsto
| r = x^r > 0
| c = Power of Positive Real Number is Positive: Integer
}}
{{eqn | o = \leadsto
| r = \sqrt [s] {\paren {x^r} } > 0
| c = Existence of Positive Root of P... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $q \in \Q$ be a [[Definition:Rational Number|rational number]].
Then:
:$x^q > 0$
where $x^q$ denotes the [[Definition:Rational Power|$x$ to the power of $q$]]. | Let $q = \dfrac r s$, where $r \in \Z$, $s \in \Z \setminus \set 0$.
Then:
{{begin-eqn}}
{{eqn | l = x > 0
| o = \leadsto
| r = x^r > 0
| c = [[Power of Positive Real Number is Positive/Integer|Power of Positive Real Number is Positive: Integer]]
}}
{{eqn | o = \leadsto
| r = \sqrt [s] {\paren ... | Power of Positive Real Number is Positive/Rational Number | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Rational_Number | https://proofwiki.org/wiki/Power_of_Positive_Real_Number_is_Positive/Rational_Number | [
"Power of Positive Real Number is Positive"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Rational Number",
"Definition:Power (Algebra)/Rational Number"
] | [
"Power of Positive Real Number is Positive/Integer",
"Existence of Positive Root of Positive Real Number",
"Category:Power of Positive Real Number is Positive"
] |
proofwiki-11489 | Power Function is Monotone/Rational Number | Let $a \in \R_{>0}$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f r = a^r$
where $a^r$ denotes $a$ to the power of $r$.
Then $f$ is monotone.
Further, $f$ is strictly monotone unless $a = 1$. | === Case 1: $a > 1$ ===
Let $a > 1$.
Then by Power Function on Base Greater than One is Strictly Increasing:
:$f$ is strictly increasing.
By Strictly Increasing Mapping is Increasing:
:$f$ is increasing. | Let $a \in \R_{>0}$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f r = a^r$
where $a^r$ denotes [[Definition:Rational Power|$a$ to the power of $r$]].
Then $f$ is [[Definition:Monotone Real Function|monotone]].
Further, $f$ is [[Definition:Strictly Monotone ... | === Case 1: $a > 1$ ===
Let $a > 1$.
Then by [[Power Function on Base Greater than One is Strictly Increasing/Rational Number|Power Function on Base Greater than One is Strictly Increasing]]:
:$f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]].
By [[Strictly Increasing Mapping is Increasin... | Power Function is Monotone/Rational Number | https://proofwiki.org/wiki/Power_Function_is_Monotone/Rational_Number | https://proofwiki.org/wiki/Power_Function_is_Monotone/Rational_Number | [
"Powers"
] | [
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Strictly Monotone/Real Function"
] | [
"Power Function on Base Greater than One is Strictly Increasing/Rational Number",
"Definition:Strictly Increasing/Real Function",
"Strictly Increasing Mapping is Increasing",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function",
"Definition:... |
proofwiki-11490 | Power Function tends to One as Power tends to Zero/Rational Number | Let $a \in \R_{> 0}$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then:
:$\ds \lim_{x \mathop \to 0} \map f x = 1$ | === Case 1: $a > 1$ ===
If $a > 1$, then:
:$\ds \lim_{x \mathop \to 0} \map f x = 1$
from Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number. | Let $a \in \R_{> 0}$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f q = a^q$
where $a^q$ denotes [[Definition:Rational Power|$a$ to the power of $q$]].
Then:
:$\ds \lim_{x \mathop \to 0} \map f x = 1$ | === Case 1: $a > 1$ ===
If $a > 1$, then:
:$\ds \lim_{x \mathop \to 0} \map f x = 1$
from [[Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number]]. | Power Function tends to One as Power tends to Zero/Rational Number | https://proofwiki.org/wiki/Power_Function_tends_to_One_as_Power_tends_to_Zero/Rational_Number | https://proofwiki.org/wiki/Power_Function_tends_to_One_as_Power_tends_to_Zero/Rational_Number | [
"Powers"
] | [
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number"
] | [
"Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number"
] |
proofwiki-11491 | Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma | Let $a \in \R$ be a real number such that $a > 1$.
Let $r \in \Q_{> 0}$ be a strictly positive rational number such that $r < 1$.
Then:
:$1 < a^r < 1 + a r$ | Define a real function $g_r: \R_{> 0} \to \R$ as:
:$\map {g_r} a = 1 + a r - a^r$
Then differentiating {{WRT|Differentiation}} $a$ gives:
:$D_a \map {g_r} a = r \paren {1 - a^{r - 1} }$
We show now that the derivative of $g_r$ is positive for all $a > 1$:
{{begin-eqn}}
{{eqn | l = r
| o = <
| r = 1
}}
{{eqn... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 1$.
Let $r \in \Q_{> 0}$ be a [[Definition:Strictly Positive Rational Number|strictly positive rational number]] such that $r < 1$.
Then:
:$1 < a^r < 1 + a r$ | Define a [[Definition:Real Function|real function]] $g_r: \R_{> 0} \to \R$ as:
:$\map {g_r} a = 1 + a r - a^r$
Then [[Definition:Differentiation|differentiating]] {{WRT|Differentiation}} $a$ gives:
:$D_a \map {g_r} a = r \paren {1 - a^{r - 1} }$
We show now that the [[Definition:Derivative|derivative]] of $g_r$ is [... | Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma | https://proofwiki.org/wiki/Power_Function_on_Base_greater_than_One_tends_to_One_as_Power_tends_to_Zero/Rational_Number/Lemma | https://proofwiki.org/wiki/Power_Function_on_Base_greater_than_One_tends_to_One_as_Power_tends_to_Zero/Rational_Number/Lemma | [
"Powers"
] | [
"Definition:Real Number",
"Definition:Strictly Positive/Rational Number"
] | [
"Definition:Real Function",
"Definition:Differentiation",
"Definition:Derivative",
"Definition:Positive Real Function",
"Power Function on Base Greater than One is Strictly Increasing/Rational Number",
"Order of Real Numbers is Dual of Order of their Negatives",
"Definition:Positive/Real Number",
"Der... |
proofwiki-11492 | Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number | Let $a \in \R_{> 0}$ be a strictly positive real number such that $a > 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f r = a^r$
where $a^r$ denotes $a$ to the power of $r$.
Then:
:$\ds \lim_{r \mathop \to 0} \map f r = 1$ | We start by treating the right-sided limit.
Let $0 < r < 1$. | Let $a \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]] such that $a > 1$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f r = a^r$
where $a^r$ denotes [[Definition:Rational Power|$a$ to the power of $r$]].
Then:
:$\... | We start by treating the [[Definition:Limit from Right|right-sided limit]].
Let $0 < r < 1$. | Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number | https://proofwiki.org/wiki/Power_Function_on_Base_greater_than_One_tends_to_One_as_Power_tends_to_Zero/Rational_Number | https://proofwiki.org/wiki/Power_Function_on_Base_greater_than_One_tends_to_One_as_Power_tends_to_Zero/Rational_Number | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number"
] | [
"Definition:Limit of Real Function/Right"
] |
proofwiki-11493 | Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number | Let $a \in \R_{> 0}$ be a strictly positive real number such that $0 < a < 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f r = a^r$
where $a^r$ denotes $a$ to the power of $r$.
Then:
:$\ds \lim_{r \mathop \to 0} \map f r = 1$ | From Ordering of Reciprocals:
:$0 < a < 1 \implies 1 < \dfrac 1 a$
So:
{{begin-eqn}}
{{eqn | l = \lim_{r \mathop \to 0} \paren {\frac 1 a}^r
| r = 1
| c = Power Function on Base greater than One tends to One as Power tends to Zero: Rational Number
}}
{{eqn | ll= \leadsto
| l = \lim_{r \mathop \to 0} \... | Let $a \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]] such that $0 < a < 1$.
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f r = a^r$
where $a^r$ denotes [[Definition:Rational Power|$a$ to the power of $r$]].
Then:... | From [[Ordering of Reciprocals]]:
:$0 < a < 1 \implies 1 < \dfrac 1 a$
So:
{{begin-eqn}}
{{eqn | l = \lim_{r \mathop \to 0} \paren {\frac 1 a}^r
| r = 1
| c = [[Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number|Power Function on Base greater than One tends to One a... | Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_Tends_to_One_as_Power_Tends_to_Zero/Rational_Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_Tends_to_One_as_Power_Tends_to_Zero/Rational_Number | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number"
] | [
"Ordering of Reciprocals",
"Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number",
"Exponent Combination Laws/Power of Quotient/Rational Numbers",
"Combination Theorem for Limits of Functions/Real/Quotient Rule",
"Category:Powers"
] |
proofwiki-11494 | Euler's Number to Rational Power permits Unique Continuous Extension | Let $e$ be Euler's number.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$f \left({q}\right) = e^q$
where $e^q$ denotes $e$ to the power of $q$.
Then there exists a unique continuous extension of $f$ to $\R$. | Since $e > 0$, we may apply Power Function to Rational Power permits Unique Continuous Extension.
Hence the result.
{{qed}}
Category:Exponential Function
j3aj4ucitdfj9juy3c0f2ti85dhfkmm | Let $e$ be [[Definition:Euler's Number|Euler's number]].
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$f \left({q}\right) = e^q$
where $e^q$ denotes [[Definition:Rational Power| $e$ to the power of $q$]].
Then there exists a [[Definition:Unique|unique]] [[Definition... | Since $e > 0$, we may apply [[Power Function to Rational Power permits Unique Continuous Extension]].
Hence the result.
{{qed}}
[[Category:Exponential Function]]
j3aj4ucitdfj9juy3c0f2ti85dhfkmm | Euler's Number to Rational Power permits Unique Continuous Extension | https://proofwiki.org/wiki/Euler's_Number_to_Rational_Power_permits_Unique_Continuous_Extension | https://proofwiki.org/wiki/Euler's_Number_to_Rational_Power_permits_Unique_Continuous_Extension | [
"Exponential Function"
] | [
"Definition:Euler's Number",
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number",
"Definition:Unique",
"Definition:Continuous Extension"
] | [
"Power Function to Rational Power permits Unique Continuous Extension",
"Category:Exponential Function"
] |
proofwiki-11495 | Open Unit Interval is Proper Subset of Closed Unit Interval | The open unit interval:
:$I_o = \openint 0 1$
is a proper subset of the closed unit interval:
:$I_c = \closedint 0 1$ | Let $x \in I_o$.
Then by definition:
:$0 < x < 1$
and so:
:$0 \le x \le 1$
and so:
:$x \in I_c$.
Thus:
:$I_o \subseteq I_c$
Consider:
:$0 \in I_c$
by definition of closed interval.
But it is not the case that $0 < 0$.
So $0 \notin I_o$ and so $I_c \nsubseteq I_o$.
Hence the result by definition of proper subset.
{{qed}... | The [[Definition:Open Unit Interval|open unit interval]]:
:$I_o = \openint 0 1$
is a [[Definition:Proper Subset|proper subset]] of the [[Definition:Closed Unit Interval|closed unit interval]]:
:$I_c = \closedint 0 1$ | Let $x \in I_o$.
Then by definition:
:$0 < x < 1$
and so:
:$0 \le x \le 1$
and so:
:$x \in I_c$.
Thus:
:$I_o \subseteq I_c$
Consider:
:$0 \in I_c$
by definition of [[Definition:Closed Real Interval|closed interval]].
But it is not the case that $0 < 0$.
So $0 \notin I_o$ and so $I_c \nsubseteq I_o$.
Hence the res... | Open Unit Interval is Proper Subset of Closed Unit Interval | https://proofwiki.org/wiki/Open_Unit_Interval_is_Proper_Subset_of_Closed_Unit_Interval | https://proofwiki.org/wiki/Open_Unit_Interval_is_Proper_Subset_of_Closed_Unit_Interval | [
"Real Intervals"
] | [
"Definition:Real Interval/Unit Interval/Open",
"Definition:Proper Subset",
"Definition:Real Interval/Unit Interval/Closed"
] | [
"Definition:Real Interval/Closed",
"Definition:Proper Subset"
] |
proofwiki-11496 | Vector Cross Product satisfies Jacobi Identity | Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in $3$ dimensional Euclidean space.
Let $\times$ denotes the cross product.
Then:
:$\mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b} = \mathbf 0$
That is, th... | {{begin-eqn}}
{{eqn | l = \mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b}
| r = \paren {\mathbf {a \cdot c} } \mathbf b - \paren {\mathbf {a \cdot b} } \mathbf c
| c =
}}
{{eqn | o =
| ... | Let $\mathbf a, \mathbf b, \mathbf c$ be [[Definition:Vector (Linear Algebra)|vectors]] in $3$ [[Definition:Dimension (Linear Algebra)|dimensional]] [[Definition:Euclidean Space|Euclidean space]].
Let $\times$ denotes the [[Definition:Cross Product|cross product]].
Then:
:$\mathbf a \times \paren {\mathbf b \times \... | {{begin-eqn}}
{{eqn | l = \mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b}
| r = \paren {\mathbf {a \cdot c} } \mathbf b - \paren {\mathbf {a \cdot b} } \mathbf c
| c =
}}
{{eqn | o =
| ... | Vector Cross Product satisfies Jacobi Identity | https://proofwiki.org/wiki/Vector_Cross_Product_satisfies_Jacobi_Identity | https://proofwiki.org/wiki/Vector_Cross_Product_satisfies_Jacobi_Identity | [
"Vector Cross Product",
"Jacobi Identity",
"Algebra"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Dimension (Linear Algebra)",
"Definition:Euclidean Space",
"Definition:Cross Product",
"Definition:Cross Product",
"Definition:Jacobi Identity"
] | [
"Lagrange's Formula",
"Dot Product Operator is Commutative",
"Category:Vector Cross Product",
"Category:Jacobi Identity",
"Category:Algebra"
] |
proofwiki-11497 | Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema | Let $\struct {S, \vee, \wedge, \preceq}$ be a meet-continuous distributive complete lattice.
Let $x \in S$.
Let $f: S \to S$ be a mapping such that
:$\forall y \in S: \map f y = x \wedge y$
Then
:$f$ preserves finite suprema | Let $X$ be finite subset of $S$ such that:
:$X$ admits a supremum
By definition of complete lattice:
:$f \sqbrk X$ admits a supremum
We will prove the result by induction on the cardinality of $X$. | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Meet-Continuous Lattice|meet-continuous]] [[Definition:Distributive Lattice|distributive]] [[Definition:Complete Lattice|complete lattice]].
Let $x \in S$.
Let $f: S \to S$ be a [[Definition:Mapping|mapping]] such that
:$\forall y \in S: \map f y = x \wedge y... | Let $X$ be [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $S$ such that:
:$X$ admits a [[Definition:Supremum of Set|supremum]]
By definition of [[Definition:Complete Lattice|complete lattice]]:
:$f \sqbrk X$ admits a [[Definition:Supremum of Set|supremum]]
We will prove the result by [[Principle ... | Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema | https://proofwiki.org/wiki/Meet-Continuous_and_Distributive_implies_Shift_Mapping_Preserves_Finite_Suprema | https://proofwiki.org/wiki/Meet-Continuous_and_Distributive_implies_Shift_Mapping_Preserves_Finite_Suprema | [
"Distributive Lattices",
"Meet-Continuous Lattices",
"Complete Lattices"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Distributive Lattice",
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Finite"
] | [
"Definition:Finite Set",
"Definition:Subset",
"Definition:Supremum of Set",
"Definition:Complete Lattice",
"Definition:Supremum of Set",
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Cardinality"
] |
proofwiki-11498 | Set Union is not Cancellable | Set union is not a cancellable operation.
That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:
:$A \cup B = A \cup C \implies B = C$ | Proof by Counterexample:
Let $S = \set {a, b}$.
Let:
:$A = \set {a, b}$
:$B = \set a$
:$C = \set b$
Then:
{{begin-eqn}}
{{eqn | l = A \cup B
| r = \set {a, b}
| c =
}}
{{eqn | r = A \cup C
| c =
}}
{{end-eqn}}
but:
{{begin-eqn}}
{{eqn | l = B
| r = \set a
| c =
}}
{{eqn | o = \ne
... | [[Definition:Set Union|Set union]] is not a [[Definition:Cancellable Operation|cancellable operation]].
That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:
:$A \cup B = A \cup C \implies B = C$ | [[Proof by Counterexample]]:
Let $S = \set {a, b}$.
Let:
:$A = \set {a, b}$
:$B = \set a$
:$C = \set b$
Then:
{{begin-eqn}}
{{eqn | l = A \cup B
| r = \set {a, b}
| c =
}}
{{eqn | r = A \cup C
| c =
}}
{{end-eqn}}
but:
{{begin-eqn}}
{{eqn | l = B
| r = \set a
| c =
}}
{{eqn | o = \... | Set Union is not Cancellable | https://proofwiki.org/wiki/Set_Union_is_not_Cancellable | https://proofwiki.org/wiki/Set_Union_is_not_Cancellable | [
"Set Union"
] | [
"Definition:Set Union",
"Definition:Cancellable Operation"
] | [
"Proof by Counterexample"
] |
proofwiki-11499 | Set Intersection is not Cancellable | Set intersection is not a cancellable operation.
That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:
:$A \cap B = A \cap C \implies B = C$ | Proof by Counterexample:
Let $S = \set {a, b, c}$.
Let:
:$A = \set a$
:$B = \set {a, b}$
:$C = \set {a, c}$
Then:
{{begin-eqn}}
{{eqn | l = A \cap B
| r = \set a
| c =
}}
{{eqn | r = A \cap C
| c =
}}
{{end-eqn}}
but:
{{begin-eqn}}
{{eqn | l = B
| r = \set {a, b}
| c =
}}
{{eqn | o = \n... | [[Definition:Set Intersection|Set intersection]] is not a [[Definition:Cancellable Operation|cancellable operation]].
That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:
:$A \cap B = A \cap C \implies B = C$ | [[Proof by Counterexample]]:
Let $S = \set {a, b, c}$.
Let:
:$A = \set a$
:$B = \set {a, b}$
:$C = \set {a, c}$
Then:
{{begin-eqn}}
{{eqn | l = A \cap B
| r = \set a
| c =
}}
{{eqn | r = A \cap C
| c =
}}
{{end-eqn}}
but:
{{begin-eqn}}
{{eqn | l = B
| r = \set {a, b}
| c =
}}
{{eqn... | Set Intersection is not Cancellable | https://proofwiki.org/wiki/Set_Intersection_is_not_Cancellable | https://proofwiki.org/wiki/Set_Intersection_is_not_Cancellable | [
"Set Intersection"
] | [
"Definition:Set Intersection",
"Definition:Cancellable Operation"
] | [
"Proof by Counterexample"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.