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-21400 | Area of Triangle Inscribed in Parabola | Let $T$ be the parabola which is
the locus of points $\tuple {x, y}$ satisfying $y = x^2$.
Let $A$ and $B$ be two arbitrary points on $T$ with coordinates:
:$A = \tuple {u, u^2}$
:$B = \tuple {v, v^2}$, with $u > v$
Let $C$ be a third point on $T$ whose $x$-coordinate is the average of those of $A$ and $B$.
The area of... | The point $\tuple {x = 0, y = 0}$ satisfies $y = x^2$.
Since $x^2 \ge 0$, $0$ is the minimum value for $y$.
Thus, the vertex of $T$ lies at the origin.
The coordinates of $C$ are given as the average those for $A$ and $B$.
Given:
:$A = \tuple {u, u^2}$
:$B = \tuple {v, v^2}$, with $u > v$
By the definition of average: ... | Let $T$ be the [[Definition:Parabola|parabola]] which is
the [[Definition:Locus|locus]] of [[Definition:Point|points]] $\tuple {x, y}$ satisfying $y = x^2$.
Let $A$ and $B$ be two arbitrary [[Definition:Point|points]] on $T$ with [[Definition:Homogeneous Cartesian Coordinates|coordinates]]:
:$A = \tuple {u, u^2}$
:$B ... | The [[Definition:Point|point]] $\tuple {x = 0, y = 0}$ satisfies $y = x^2$.
Since $x^2 \ge 0$, $0$ is the [[Definition:Minimum Value of Real Function|minimum value]] for $y$.
Thus, the [[Definition:Parabola|vertex]] of $T$ lies at the [[Definition:Origin of Coordinates|origin]].
The [[Definition:Homogeneous Cartesia... | Area of Triangle Inscribed in Parabola/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola/Proof_1 | [
"Area of Triangle Inscribed in Parabola",
"Triangles",
"Parabolas",
"Areas of Triangles"
] | [
"Definition:Parabola",
"Definition:Locus",
"Definition:Point",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Arithmetic Mean",
"Definition:Area"
] | [
"Definition:Point",
"Definition:Minimum Value of Real Function",
"Definition:Parabola",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Arithmetic Mean",
"Definition:Arithmetic Mean",
"Definition:Line/Midpoint",
"Definition:Homogeneous Cartesian Coo... |
proofwiki-21401 | Area of Triangle Inscribed in Parabola | Let $T$ be the parabola which is
the locus of points $\tuple {x, y}$ satisfying $y = x^2$.
Let $A$ and $B$ be two arbitrary points on $T$ with coordinates:
:$A = \tuple {u, u^2}$
:$B = \tuple {v, v^2}$, with $u > v$
Let $C$ be a third point on $T$ whose $x$-coordinate is the average of those of $A$ and $B$.
The area of... | From Two-Point Form of Equation of Straight Line in Plane, the straight line $AB$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \dfrac {\paren {y_{AB} - y_1} } {\paren {x - x_1} }
| r = \dfrac {\paren {y_2 - y_1} } {\paren {x_2 - x_1} }
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {y_{AB} - v... | Let $T$ be the [[Definition:Parabola|parabola]] which is
the [[Definition:Locus|locus]] of [[Definition:Point|points]] $\tuple {x, y}$ satisfying $y = x^2$.
Let $A$ and $B$ be two arbitrary [[Definition:Point|points]] on $T$ with [[Definition:Homogeneous Cartesian Coordinates|coordinates]]:
:$A = \tuple {u, u^2}$
:$B ... | From [[Two-Point Form of Equation of Straight Line in Plane]], the [[Definition:Straight Line|straight line]] $AB$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \dfrac {\paren {y_{AB} - y_1} } {\paren {x - x_1} }
| r = \dfrac {\paren {y_2 - y_1} } {\paren {x_2 - x_1} }
| c =
}}
{{eqn | ll= \leadsto
... | Area of Triangle Inscribed in Parabola/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola/Proof_2 | [
"Area of Triangle Inscribed in Parabola",
"Triangles",
"Parabolas",
"Areas of Triangles"
] | [
"Definition:Parabola",
"Definition:Locus",
"Definition:Point",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Arithmetic Mean",
"Definition:Area"
] | [
"Equation of Straight Line in Plane/Two-Point Form",
"Definition:Line/Straight Line",
"Difference of Two Squares",
"Definition:Division/Field/Real Numbers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Line/Midpoint",
... |
proofwiki-21402 | Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:$\struct {\map {C^*} {S, ... | From Ring of Continuous Real-Valued Functions is Ring:
:$\struct {\map C {S, \R}, +, *}$ is a ring.
From Additive Inverse in Ring of Continuous Real-Valued Functions:
:$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
::$\forall s \in S: \map {\paren {-f} } s := - \map f s$
F... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Let $\struct {\... | From [[Ring of Continuous Real-Valued Functions is Ring]]:
:$\struct {\map C {S, \R}, +, *}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
From [[Additive Inverse in Ring of Continuous Real-Valued Functions]]:
:$\forall f \in R^S :$ the [[Definition:Additive Inverse in Ring|additive inverse]] of $f$ is the [[Defi... | Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Subring_of_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Subring_of_Continuous_Real-Valued_Functions | [
"Rings of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Bounded Continuous Real-Valued Functions",
"Definition:Subring"
] | [
"Ring of Continuous Real-Valued Functions is Ring",
"Definition:Ring (Abstract Algebra)",
"Additive Inverse in Ring of Continuous Real-Valued Functions",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function",
"Subring Test",
"Definition:Subring",
"Subring Test",
... |
proofwiki-21403 | Zero of Ring of Bounded Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:the zero of $\struct{\map {C^*} {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to \R$ defined by:
::$\forall s \... | Let $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a subring of $\struct{\map C {S, \R}, +, *}$
From Zero of Ring of Continuous Real-Valued Functio... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the [[Definition:Ring of Bounded Continuous Real-Valued Functions|ring of bounded continuous real-valued functions]] from $... | Let $\struct{\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions from $S$]].
From [[Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions]]:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a [[Definition:Subring|subring]] of... | Zero of Ring of Bounded Continuous Real-Valued Functions | https://proofwiki.org/wiki/Zero_of_Ring_of_Bounded_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Zero_of_Ring_of_Bounded_Continuous_Real-Valued_Functions | [
"Rings of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Bounded Continuous Real-Valued Functions",
"Definition:Ring Zero",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions",
"Definition:Subring",
"Zero of Ring of Continuous Real-Valued Functions",
"Definition:Ring Zero",
"Definition:Constant Mapping",
"Zero of Subring is Zero of Ring"... |
proofwiki-21404 | Unity of Ring of Bounded Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:the unity of $\struct{\map {C^*} {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:
::$\forall s ... | Let $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a subring of $\struct{\map C {S, \R}, +, *}$
From Unity of Ring of Continuous Real-Valued Functi... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the [[Definition:Ring of Bounded Continuous Real-Valued Functions|ring of bounded continuous real-valued functions]] from $... | Let $\struct{\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions from $S$]].
From [[Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions]]:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a [[Definition:Subring|subring]] of... | Unity of Ring of Bounded Continuous Real-Valued Functions | https://proofwiki.org/wiki/Unity_of_Ring_of_Bounded_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Unity_of_Ring_of_Bounded_Continuous_Real-Valued_Functions | [
"Rings of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Bounded Continuous Real-Valued Functions",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions",
"Definition:Subring",
"Unity of Ring of Continuous Real-Valued Functions",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping",
"Constant Ma... |
proofwiki-21405 | Ring of Bounded Continuous Real-Valued Functions is Commutative | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a commutative ring | Let $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a subring of $\struct{\map C {S, \R}, +, *}$
From Ring of Continuous Real-Valued Functions is Co... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the [[Definition:Ring of Bounded Continuous Real-Valued Functions|ring of bounded continuous real-valued functions]] from $... | Let $\struct{\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions from $S$]].
From [[Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions]]:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a [[Definition:Subring|subring]] of... | Ring of Bounded Continuous Real-Valued Functions is Commutative | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Real-Valued_Functions_is_Commutative | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Real-Valued_Functions_is_Commutative | [
"Rings of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Bounded Continuous Real-Valued Functions",
"Definition:Commutative Ring"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions",
"Definition:Subring",
"Ring of Continuous Real-Valued Functions is Commutative",
"Definition:Commutative Ring",
"Subring of Commutative Ring is Commutative",
"Def... |
proofwiki-21406 | Zero of Ring of Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Then:
:the zero of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to \R$ defined by:
::$\forall s \in S : \map {0_{... | By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Zero of Ring of Continuous Mappings:
:the zero of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to \R$ defined by:
::$\forall s \in S : \map {0_... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Then:
:the [[D... | By definition of [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]]:
:$\struct {\map C {S, \R}, +, *}$ is the [[Definition:Ring of Continuous Mappings|ring of continuous mappings]] from $S$ to $\R$.
From [[Zero of Ring of Continuous Mappings]]:
:the [[Definition:Ring Zero|... | Zero of Ring of Continuous Real-Valued Functions | https://proofwiki.org/wiki/Zero_of_Ring_of_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Zero_of_Ring_of_Continuous_Real-Valued_Functions | [
"Rings of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring Zero",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Continuous Mappings",
"Zero of Ring of Continuous Mappings",
"Definition:Ring Zero",
"Definition:Constant Mapping"
] |
proofwiki-21407 | Unity of Ring of Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Then:
:the unity of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:
::$\forall s \in S : \map {1_... | By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Unity of Ring of Continuous Mappings:
:the unity of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:
::$\forall s \in S : \map {... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Then:
:the [[D... | By definition of [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]]:
:$\struct {\map C {S, \R}, +, *}$ is the [[Definition:Ring of Continuous Mappings|ring of continuous mappings]] from $S$ to $\R$.
From [[Unity of Ring of Continuous Mappings]]:
:the [[Definition:Unity of ... | Unity of Ring of Continuous Real-Valued Functions | https://proofwiki.org/wiki/Unity_of_Ring_of_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Unity_of_Ring_of_Continuous_Real-Valued_Functions | [
"Rings of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Continuous Mappings",
"Unity of Ring of Continuous Mappings",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping"
] |
proofwiki-21408 | Ring of Continuous Real-Valued Functions is Commutative | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Then:
:$\struct{\map C {S, \R}, +, *}$ is a commutative ring | By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Commutativity of Ring of Continuous Mappings:
:$\struct{\map C {S, \R}, +, *}$ is a commutative ring
{{qed}} | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Then:
:$\struc... | By definition of [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]]:
:$\struct {\map C {S, \R}, +, *}$ is the [[Definition:Ring of Continuous Mappings|ring of continuous mappings]] from $S$ to $\R$.
From [[Commutativity of Ring of Continuous Mappings]]:
:$\struct{\map C {S... | Ring of Continuous Real-Valued Functions is Commutative | https://proofwiki.org/wiki/Ring_of_Continuous_Real-Valued_Functions_is_Commutative | https://proofwiki.org/wiki/Ring_of_Continuous_Real-Valued_Functions_is_Commutative | [
"Rings of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Commutative Ring"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Continuous Mappings",
"Commutativity of Ring of Continuous Mappings",
"Definition:Commutative Ring"
] |
proofwiki-21409 | Ring of Continuous Real-Valued Functions is Ring | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Then:
:$\struct {\map C {S, \R}, +, *}$ is a ring | By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct {\map C {S, \R}, +, *}$ is a subring
By definition of subring:
:$\struct {\map C {S, \R}, +, *}$ is a r... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Then:
:$\struc... | By definition of [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]]:
:$\struct {\map C {S, \R}, +, *}$ is the [[Definition:Ring of Continuous Mappings|ring of continuous mappings]] from $S$ to $\R$.
From [[Ring of Continuous Mappings is Subring of All Mappings]]:
:$\struct... | Ring of Continuous Real-Valued Functions is Ring | https://proofwiki.org/wiki/Ring_of_Continuous_Real-Valued_Functions_is_Ring | https://proofwiki.org/wiki/Ring_of_Continuous_Real-Valued_Functions_is_Ring | [
"Rings of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Continuous Mappings",
"Ring of Continuous Mappings is Subring of All Mappings",
"Definition:Subring",
"Definition:Subring",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-21410 | Cone on Compact Space is Compact | Let $A$ be a compact topological space.
Let $C A$ denote the cone on $A$.
Then, $C A$ is compact. | By definition of cone:
:$C A = T \ast A$
where:
:$T$ denotes the trivial topological space
:$T \ast A$ denotes the join of $T$ and $A$
By Finite Topological Space is Compact, $T$ is compact.
Therefore, by Join of Compact Spaces is Compact:
:$C A$ is compact.
{{qed}}
Category:Compact Topological Spaces
jmdxk0frvkvoqotn5... | Let $A$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $C A$ denote the [[Definition:Cone (Topology)|cone]] on $A$.
Then, $C A$ is [[Definition:Compact Topological Space|compact]]. | By definition of [[Definition:Cone (Topology)|cone]]:
:$C A = T \ast A$
where:
:$T$ denotes the [[Definition:Trivial Topological Space|trivial topological space]]
:$T \ast A$ denotes the [[Definition:Join (Topology)|join]] of $T$ and $A$
By [[Finite Topological Space is Compact]], $T$ is [[Definition:Compact Topologic... | Cone on Compact Space is Compact | https://proofwiki.org/wiki/Cone_on_Compact_Space_is_Compact | https://proofwiki.org/wiki/Cone_on_Compact_Space_is_Compact | [
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Cone (Topology)",
"Definition:Compact Topological Space"
] | [
"Definition:Cone (Topology)",
"Definition:Trivial Topological Space",
"Definition:Join (Topology)",
"Finite Topological Space is Compact",
"Definition:Compact Topological Space",
"Join of Compact Spaces is Compact",
"Definition:Compact Topological Space",
"Category:Compact Topological Spaces"
] |
proofwiki-21411 | Euler Phi Function of Product | Let $m$ and $n$ be positive integers.
Let $d$ be the greatest common divisor of $m$ and $n$.
Then:
:$\map \phi {m n} = \map \phi m \map \phi n \paren {\dfrac d {\map \phi d} }$
where $\phi$ is the Euler $\phi$ function. | From Euler Phi Function of Integer, we have:
:$\ds \frac {\map \phi m} m = \prod_{p \mathop \divides m} \paren {1 - \frac 1 p}$
:$\ds \frac {\map \phi n} n = \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
:$\ds \frac {\map \phi {m n} } {m n} = \prod_{p \mathop \divides m n} \paren {1 - \frac 1 p}$
where the produ... | Let $m$ and $n$ be [[Definition:Positive Integer|positive integers]].
Let $d$ be the [[Definition:Greatest Common Divisor|greatest common divisor]] of $m$ and $n$.
Then:
:$\map \phi {m n} = \map \phi m \map \phi n \paren {\dfrac d {\map \phi d} }$
where $\phi$ is the [[Definition:Euler Phi Function|Euler $\phi$ func... | From [[Euler Phi Function of Integer]], we have:
:$\ds \frac {\map \phi m} m = \prod_{p \mathop \divides m} \paren {1 - \frac 1 p}$
:$\ds \frac {\map \phi n} n = \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
:$\ds \frac {\map \phi {m n} } {m n} = \prod_{p \mathop \divides m n} \paren {1 - \frac 1 p}$
where the [... | Euler Phi Function of Product | https://proofwiki.org/wiki/Euler_Phi_Function_of_Product | https://proofwiki.org/wiki/Euler_Phi_Function_of_Product | [
"Euler Phi Function"
] | [
"Definition:Positive/Integer",
"Definition:Greatest Common Divisor",
"Definition:Euler Phi Function"
] | [
"Euler Phi Function of Integer",
"Definition:Product",
"Definition:Prime Number",
"Euclid's Lemma for Prime Divisors",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Inte... |
proofwiki-21412 | Parallel Chords Cut Equal Chords in a Circle | Let $\CC$ be a circle on center $O$.
Let $AB$ and $CD$ be chords in $\CC$ with $AB \parallel CD$.
Then the two chords cut from the circle by $AB$ and $CD$ are equal:
:$AC = BD$ | 300px
Draw $AD$.
By Parallelism implies Equal Alternate Angles:
:$\angle BAD = \angle ADC$
The central angles corresponding to $\angle BAD$ and $\angle ADC$ are equal by Inscribed Angle Theorem:
:$\angle BOD = \angle AOC$
As radii of $\CC$:
:$OB = OD = OC = OA$
By Triangle Side-Angle-Side Congruence:
:$\triangle OAC \c... | Let $\CC$ be a [[Definition:Circle|circle]] on [[Definition:Center of Circle|center]] $O$.
Let $AB$ and $CD$ be [[Definition:Chord of Circle|chords]] in $\CC$ with $AB \parallel CD$.
Then the two [[Definition:Chord of Circle|chords]] cut from the circle by $AB$ and $CD$ are equal:
:$AC = BD$ | [[File:Chords 6.png|300px]]
Draw $AD$.
By [[Parallelism implies Equal Alternate Angles]]:
:$\angle BAD = \angle ADC$
The [[Definition:Central Angle|central angles]] corresponding to $\angle BAD$ and $\angle ADC$ are equal by [[Inscribed Angle Theorem]]:
:$\angle BOD = \angle AOC$
As [[Definition:Radius of Circle|ra... | Parallel Chords Cut Equal Chords in a Circle | https://proofwiki.org/wiki/Parallel_Chords_Cut_Equal_Chords_in_a_Circle | https://proofwiki.org/wiki/Parallel_Chords_Cut_Equal_Chords_in_a_Circle | [
"Circles",
"Parallel Lines"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Chord",
"Definition:Circle/Chord"
] | [
"File:Chords 6.png",
"Parallelism implies Equal Alternate Angles",
"Definition:Sector of Circle/Angle",
"Inscribed Angle Theorem",
"Definition:Circle/Radius",
"Triangle Side-Angle-Side Congruence",
"Category:Circles",
"Category:Parallel Lines"
] |
proofwiki-21413 | Cofunction of Cofunction | Let $g$ be a cofunction of $f$.
Then $f$ is a cofunction of $g$.
That is, cofunctions exist in pairs. | That is:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map g x
| r = \map f {90 \degrees - x}
| c = {{Defof|Cofunction}}
}}
{{eqn | l = \map g {90 \degrees - x}
| r = \map f {90 \degrees - \paren {90 \degrees - x} }
| c =
}}
{{eqn | r = \map f x
| c =
}}
{{end-eqn}}
Hence the r... | Let $g$ be a [[Definition:Cofunction|cofunction]] of $f$.
Then $f$ is a [[Definition:Cofunction|cofunction]] of $g$.
That is, [[Definition:Cofunction|cofunctions]] exist in [[Definition:Doubleton|pairs]]. | That is:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map g x
| r = \map f {90 \degrees - x}
| c = {{Defof|Cofunction}}
}}
{{eqn | l = \map g {90 \degrees - x}
| r = \map f {90 \degrees - \paren {90 \degrees - x} }
| c =
}}
{{eqn | r = \map f x
| c =
}}
{{end-eqn}}
Hence the ... | Cofunction of Cofunction | https://proofwiki.org/wiki/Cofunction_of_Cofunction | https://proofwiki.org/wiki/Cofunction_of_Cofunction | [
"Cofunctions"
] | [
"Definition:Cofunction",
"Definition:Cofunction",
"Definition:Cofunction",
"Definition:Doubleton"
] | [
"Definition:Cofunction"
] |
proofwiki-21414 | Sine and Cosine are Cofunctions | The sine and cosine are cofunctions:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \sin x
| r = \map \cos {90 \degrees - x}
}}
{{eqn | l = \cos x
| r = \map \sin {90 \degrees - x}
}}
{{end-eqn}} | We have:
:Sine of Complement equals Cosine
:Cosine of Complement equals Sine
Hence the result by definition of cofunction.
{{qed}} | The [[Definition:Real Sine Function|sine]] and [[Definition:Real Cosine Function|cosine]] are [[Definition:Cofunction|cofunctions]]:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \sin x
| r = \map \cos {90 \degrees - x}
}}
{{eqn | l = \cos x
| r = \map \sin {90 \degrees - x}
}}
{{end-eqn}} | We have:
:[[Sine of Complement equals Cosine]]
:[[Cosine of Complement equals Sine]]
Hence the result by definition of [[Definition:Cofunction|cofunction]].
{{qed}} | Sine and Cosine are Cofunctions | https://proofwiki.org/wiki/Sine_and_Cosine_are_Cofunctions | https://proofwiki.org/wiki/Sine_and_Cosine_are_Cofunctions | [
"Cofunctions"
] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Cofunction"
] | [
"Sine of Complement equals Cosine",
"Cosine of Complement equals Sine",
"Definition:Cofunction"
] |
proofwiki-21415 | Tangent and Cotangent are Cofunctions | The tangent and cotangent are cofunctions:
{{begin-eqn}}
{{eqn | n = 1
| q = \forall x \in \R, \cos x \ne 0
| l = \tan x
| r = \map \cot {90 \degrees - x}
}}
{{eqn | n = 2
| q = \forall x \in \R, \sin x \ne 0
| l = \cot x
| r = \map \tan {90 \degrees - x}
}}
{{end-eqn}} | === Proof of $(1)$ ===
From Tangent is Sine divided by Cosine:
:$\tan x = \dfrac {\sin x} {\cos x}$
Hence in order for $\tan x$ to be defined it is necessary for $\cos x \ne 0$.
Then we have:
:Tangent of Complement equals Cotangent
{{qed|lemma}} | The [[Definition:Real Tangent Function|tangent]] and [[Definition:Real Cotangent Function|cotangent]] are [[Definition:Cofunction|cofunctions]]:
{{begin-eqn}}
{{eqn | n = 1
| q = \forall x \in \R, \cos x \ne 0
| l = \tan x
| r = \map \cot {90 \degrees - x}
}}
{{eqn | n = 2
| q = \forall x \in \... | === Proof of $(1)$ ===
From [[Tangent is Sine divided by Cosine]]:
:$\tan x = \dfrac {\sin x} {\cos x}$
Hence in order for $\tan x$ to be defined it is necessary for $\cos x \ne 0$.
Then we have:
:[[Tangent of Complement equals Cotangent]]
{{qed|lemma}} | Tangent and Cotangent are Cofunctions | https://proofwiki.org/wiki/Tangent_and_Cotangent_are_Cofunctions | https://proofwiki.org/wiki/Tangent_and_Cotangent_are_Cofunctions | [
"Cofunctions"
] | [
"Definition:Tangent Function/Real",
"Definition:Cotangent/Real Function",
"Definition:Cofunction"
] | [
"Tangent is Sine divided by Cosine",
"Tangent of Complement equals Cotangent"
] |
proofwiki-21416 | Secant and Cosecant are Cofunctions | The secant and cosecant are cofunctions:
{{begin-eqn}}
{{eqn | n = 1
| q = \forall x \in \R, \cos x \ne 0
| l = \sec x
| r = \map \csc {90 \degrees - x}
}}
{{eqn | n = 2
| q = \forall x \in \R, \sin x \ne 0
| l = \csc x
| r = \map \sec {90 \degrees - x}
}}
{{end-eqn}} | === Proof of $(1)$ ===
From Secant is Reciprocal of Cosine:
:$\sec x = \dfrac 1 {\cos x}$
Hence in order for $\sec x$ to be defined it is necessary for $\cos x \ne 0$.
Then we have:
:Secant of Complement equals Cosecant
{{qed|lemma}} | The [[Definition:Real Secant Function|secant]] and [[Definition:Real Cosecant Function|cosecant]] are [[Definition:Cofunction|cofunctions]]:
{{begin-eqn}}
{{eqn | n = 1
| q = \forall x \in \R, \cos x \ne 0
| l = \sec x
| r = \map \csc {90 \degrees - x}
}}
{{eqn | n = 2
| q = \forall x \in \R, \... | === Proof of $(1)$ ===
From [[Secant is Reciprocal of Cosine]]:
:$\sec x = \dfrac 1 {\cos x}$
Hence in order for $\sec x$ to be defined it is necessary for $\cos x \ne 0$.
Then we have:
:[[Secant of Complement equals Cosecant]]
{{qed|lemma}} | Secant and Cosecant are Cofunctions | https://proofwiki.org/wiki/Secant_and_Cosecant_are_Cofunctions | https://proofwiki.org/wiki/Secant_and_Cosecant_are_Cofunctions | [
"Cofunctions"
] | [
"Definition:Secant Function/Real",
"Definition:Cosecant/Real Function",
"Definition:Cofunction"
] | [
"Secant is Reciprocal of Cosine",
"Secant of Complement equals Cosecant"
] |
proofwiki-21417 | Kinetic Energy is not necessarily Conserved in a Collision | Let two bodies be in collision.
The total kinetic energy of the two bodies before the collision may not necessarily be the same as the total kinetic energy of the two bodies after the collision. | {{ProofWanted|some background work needed here}} | Let two [[Definition:Body|bodies]] be in [[Definition:Collision|collision]].
The total [[Definition:Kinetic Energy|kinetic energy]] of the two [[Definition:Body|bodies]] before the [[Definition:Collision|collision]] may not necessarily be the same as the total [[Definition:Kinetic Energy|kinetic energy]] of the two [[... | {{ProofWanted|some background work needed here}} | Kinetic Energy is not necessarily Conserved in a Collision | https://proofwiki.org/wiki/Kinetic_Energy_is_not_necessarily_Conserved_in_a_Collision | https://proofwiki.org/wiki/Kinetic_Energy_is_not_necessarily_Conserved_in_a_Collision | [
"Collisions",
"Kinetic Energy"
] | [
"Definition:Body",
"Definition:Collision",
"Definition:Kinetic Energy",
"Definition:Body",
"Definition:Collision",
"Definition:Kinetic Energy",
"Definition:Body",
"Definition:Collision"
] | [] |
proofwiki-21418 | Column Rank of Matrix equals Row Rank | Let $\mathbf A$ be a matrix.
The column rank of $\mathbf A$ is equal to the row rank of $\mathbf A$. | === Proof Outline and Definitions ===
Recall:
:The row rank of $\mathbf A$ is defined as the dimension of the row space.
:The column rank of $\mathbf A$ is defined as the dimension of the column space.
{{WLOG}}, we define the rank of $\mathbf A$ to be the column rank of $\mathbf A$.
{{ExtractTheorem|We announce and pro... | Let $\mathbf A$ be a [[Definition:Matrix|matrix]].
The [[Definition:Column Rank|column rank]] of $\mathbf A$ is equal to the [[Definition:Row Rank|row rank]] of $\mathbf A$. | === Proof Outline and Definitions ===
Recall:
:The [[Definition:Row Rank|row rank]] of $\mathbf A$ is defined as the [[Definition:Dimension (Linear Algebra)|dimension]] of the [[Definition:Row Space|row space]].
:The [[Definition:Column Rank|column rank]] of $\mathbf A$ is defined as the [[Definition:Dimension (Line... | Column Rank of Matrix equals Row Rank | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank | [
"Column Rank of Matrix equals Row Rank",
"Column Rank",
"Row Rank"
] | [
"Definition:Matrix",
"Definition:Column Rank",
"Definition:Row Rank"
] | [
"Definition:Row Rank",
"Definition:Dimension (Linear Algebra)",
"Definition:Row Space",
"Definition:Column Rank",
"Definition:Dimension (Linear Algebra)",
"Definition:Column Space",
"Definition:Rank/Matrix",
"Definition:Column Rank",
"Row Rank of Transpose is Column Rank",
"Column Rank of Transpos... |
proofwiki-21419 | Column Rank of Matrix equals Row Rank | Let $\mathbf A$ be a matrix.
The column rank of $\mathbf A$ is equal to the row rank of $\mathbf A$. | Let $\map c {\mathbf A}$ denote the column rank of $\mathbf A$.
Let $\map r {\mathbf A}$ denote the row rank of $\mathbf A$.
Let $\mathbf A$ be an $m\times n$ matrix whose row rank is $r$.
Therefore, the dimension of the row space of $\mathbf A$ is $r$.
Let $\mathbf x_1, \ldots, \mathbf x_r$ be a basis of the row space... | Let $\mathbf A$ be a [[Definition:Matrix|matrix]].
The [[Definition:Column Rank|column rank]] of $\mathbf A$ is equal to the [[Definition:Row Rank|row rank]] of $\mathbf A$. | Let $\map c {\mathbf A}$ denote the [[Definition:Column Rank|column rank]] of $\mathbf A$.
Let $\map r {\mathbf A}$ denote the [[Definition:Row Rank|row rank]] of $\mathbf A$.
Let $\mathbf A$ be an $m\times n$ [[Definition:Matrix|matrix]] whose [[Definition:Row Rank|row rank]] is $r$.
Therefore, the [[Definition:D... | Column Rank of Matrix equals Row Rank/Proof using Orthogonality | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank/Proof_using_Orthogonality | [
"Column Rank of Matrix equals Row Rank",
"Column Rank",
"Row Rank"
] | [
"Definition:Matrix",
"Definition:Column Rank",
"Definition:Row Rank"
] | [
"Definition:Column Rank",
"Definition:Row Rank",
"Definition:Matrix",
"Definition:Row Rank",
"Definition:Dimension (Linear Algebra)",
"Definition:Row Space",
"Definition:Row Space",
"Definition:Row Space",
"Definition:Row Space",
"Definition:Row Space",
"Definition:Row Space",
"Definition:Colu... |
proofwiki-21420 | Column Rank of Matrix equals Row Rank | Let $\mathbf A$ be a matrix.
The column rank of $\mathbf A$ is equal to the row rank of $\mathbf A$. | Let $\map c {\mathbf A}$ denote the column rank of $\mathbf A$.
Let $\map r {\mathbf A}$ denote the row rank of $\mathbf A$.
Let $\mathbf A$ be an $m \times n$ matrix whose column rank is $r$.
Therefore, the dimension of the column space of $\mathbf A$ is $r$.
Let $\mathbf c_1, \ldots, \mathbf c_r$ be any basis for the... | Let $\mathbf A$ be a [[Definition:Matrix|matrix]].
The [[Definition:Column Rank|column rank]] of $\mathbf A$ is equal to the [[Definition:Row Rank|row rank]] of $\mathbf A$. | Let $\map c {\mathbf A}$ denote the [[Definition:Column Rank|column rank]] of $\mathbf A$.
Let $\map r {\mathbf A}$ denote the [[Definition:Row Rank|row rank]] of $\mathbf A$.
Let $\mathbf A$ be an $m \times n$ [[Definition:Matrix|matrix]] whose [[Definition:Column Rank|column rank]] is $r$.
Therefore, the [[Defin... | Column Rank of Matrix equals Row Rank/Proof using Rank Factorization | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Row_Rank/Proof_using_Rank_Factorization | [
"Column Rank of Matrix equals Row Rank",
"Column Rank",
"Row Rank"
] | [
"Definition:Matrix",
"Definition:Column Rank",
"Definition:Row Rank"
] | [
"Definition:Column Rank",
"Definition:Row Rank",
"Definition:Matrix",
"Definition:Column Rank",
"Definition:Dimension (Linear Algebra)",
"Definition:Column Space",
"Definition:Column Space",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Column ... |
proofwiki-21421 | Column Rank of Matrix equals Rank of Matrix | Let $\mathbf A$ be a matrix.
The column rank of $\mathbf A$ is equal to the rank of $\mathbf A$. | The rank of $\mathbf A$ is defined as the dimension of the column space of $\mathbf A$.
Thus the result follows by definition.
{{qed}} | Let $\mathbf A$ be a [[Definition:Matrix|matrix]].
The [[Definition:Column Rank|column rank]] of $\mathbf A$ is equal to the [[Definition:Rank of Matrix|rank]] of $\mathbf A$. | The [[Definition:Rank of Matrix|rank]] of $\mathbf A$ is defined as the [[Definition:Dimension of Vector Space|dimension]] of the [[Definition:Column Space|column space]] of $\mathbf A$.
Thus the result follows by definition.
{{qed}} | Column Rank of Matrix equals Rank of Matrix | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Rank_of_Matrix | https://proofwiki.org/wiki/Column_Rank_of_Matrix_equals_Rank_of_Matrix | [
"Column Rank",
"Rank of Matrix"
] | [
"Definition:Matrix",
"Definition:Column Rank",
"Definition:Rank/Matrix"
] | [
"Definition:Rank/Matrix",
"Definition:Dimension of Vector Space",
"Definition:Column Space"
] |
proofwiki-21422 | Constant Real-Valued Function is Bounded | Let $S$ be a set.
Let $\R$ denote the real number line.
Let $c \in \R$.
Let $c_{\R^S} : S \to R$ be the constant mapping defined by:
:$\forall s \in S : \map {c_{\R^S} } s = c$
Then $c_{\R^S}$ is a bounded real-valued function. | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {c_{\R^S} } s}
| r = \size c
| c = {{Defof|Constant Mapping}}
}}
{{eqn | o = \le
| r = \size c
}}
{{end-eqn}}
It follows that $c_{\R^S}$ is a bounded real-valued function by definition.
{{qed}}
Category:Constant Mappings
Categ... | Let $S$ be a [[Definition:Set|set]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $c \in \R$.
Let $c_{\R^S} : S \to R$ be the [[Definition:Constant Mapping|constant mapping]] defined by:
:$\forall s \in S : \map {c_{\R^S} } s = c$
Then $c_{\R^S}$ is a [[Definition:Bounded Real-Valued F... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {c_{\R^S} } s}
| r = \size c
| c = {{Defof|Constant Mapping}}
}}
{{eqn | o = \le
| r = \size c
}}
{{end-eqn}}
It follows that $c_{\R^S}$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] by definit... | Constant Real-Valued Function is Bounded | https://proofwiki.org/wiki/Constant_Real-Valued_Function_is_Bounded | https://proofwiki.org/wiki/Constant_Real-Valued_Function_is_Bounded | [
"Constant Mappings",
"Bounded Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Real Number/Real Number Line",
"Definition:Constant Mapping",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Category:Constant Mappings",
"Category:Bounded Real-Valued Functions"
] |
proofwiki-21423 | Refinement of Open Cover has Greater Entropy | Let $X$ be a compact topological space.
Let $\alpha, \beta$ be open covers of $X$.
Let $\map H \alpha$ and $\map H \beta$ be their entropies.
Suppose that $\beta$ is a refinement of $\alpha$.
Then:
:$\map H \alpha \le \map H \beta$ | By definition of entropy of open cover, there is a finite subcover $\beta' \subseteq \beta$ such that:
:$\map H \beta = \map \ln k$
with $k = \size {\beta '}$.
We write:
:$\beta' = \set {B_1, \ldots, B_k}$
{{Recall|Refinement of Cover}}
:$\forall i \in \set {1, \ldots ,k} \; \exists A_i \in \alpha : B_i \subseteq A_i$
... | Let $X$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\alpha, \beta$ be [[Definition:Open Cover|open covers]] of $X$.
Let $\map H \alpha$ and $\map H \beta$ be their [[Definition:Entropy of Open Cover|entropies]].
Suppose that $\beta$ is a [[Definition:Refinement of Cover|refinement]]... | By definition of [[Definition:Entropy of Open Cover|entropy of open cover]], there is a [[Definition:Finite Cover|finite]] [[Definition:Subcover|subcover]] $\beta' \subseteq \beta$ such that:
:$\map H \beta = \map \ln k$
with $k = \size {\beta '}$.
We write:
:$\beta' = \set {B_1, \ldots, B_k}$
{{Recall|Refinement of ... | Refinement of Open Cover has Greater Entropy | https://proofwiki.org/wiki/Refinement_of_Open_Cover_has_Greater_Entropy | https://proofwiki.org/wiki/Refinement_of_Open_Cover_has_Greater_Entropy | [
"Ergodic Theory",
"Topology"
] | [
"Definition:Compact Topological Space",
"Definition:Open Cover",
"Definition:Entropy of Open Cover",
"Definition:Refinement of Cover"
] | [
"Definition:Entropy of Open Cover",
"Definition:Cover of Set/Finite",
"Definition:Subcover",
"Definition:Subcover",
"Definition:Entropy of Open Cover"
] |
proofwiki-21424 | Vector Magnitude is Invariant Under Rotation | Let $\mathbf v$ be an arbitrary vector in the Cartesian plane $\CC$.
Let the coordinate system then be rotated in the anticlockwise direction by an arbitrary angle $\theta$.
Then:
the magnitude of $\mathbf v$ is unchanged in the new coordinate system. | Let $P = \tuple {x_1, y_1}$ be the initial point of $\mathbf v$.
Let $Q = \tuple {x_2, y_2}$ be the terminal point of $\mathbf v$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v
| r = \tuple {X, Y}
| c = {{Defof|Vector Quantity}}
}}
{{eqn | l = \tuple {X, Y}
| r = \tuple {x_2 - x_1, y_2 - y_1}
| c = ... | Let $\mathbf v$ be an arbitrary [[Definition:Vector Quantity|vector]] in the [[Definition:Cartesian Coordinate System|Cartesian plane]] $\CC$.
Let the [[Definition:Cartesian Coordinate System|coordinate system]] then be [[Definition:Rotation (Geometry)|rotated]] in the [[Definition:Anticlockwise|anticlockwise directio... | Let $P = \tuple {x_1, y_1}$ be the [[Definition:Initial Point of Vector|initial point]] of $\mathbf v$.
Let $Q = \tuple {x_2, y_2}$ be the [[Definition:Terminal Point of Vector|terminal point]] of $\mathbf v$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v
| r = \tuple {X, Y}
| c = {{Defof|Vector Quantity}}
}... | Vector Magnitude is Invariant Under Rotation/Proof 1 | https://proofwiki.org/wiki/Vector_Magnitude_is_Invariant_Under_Rotation | https://proofwiki.org/wiki/Vector_Magnitude_is_Invariant_Under_Rotation/Proof_1 | [
"Vector Magnitude is Invariant Under Rotation",
"Geometric Rotations",
"Vector Length"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Rotation (Geometry)",
"Definition:Anticlockwise",
"Definition:Angle",
"Definition:Magnitude",
"Definition:Cartesian Coordinate System"
] | [
"Definition:Initial Point of Vector",
"Definition:Terminal Point of Vector",
"Definition:Magnitude",
"Definition:Square",
"Equations defining Plane Rotation",
"Definition:Vector Quantity",
"Definition:Rotation (Geometry)",
"Definition:Cartesian Coordinate System",
"Definition:Vector Quantity/Compone... |
proofwiki-21425 | Vector Magnitude is Invariant Under Rotation | Let $\mathbf v$ be an arbitrary vector in the Cartesian plane $\CC$.
Let the coordinate system then be rotated in the anticlockwise direction by an arbitrary angle $\theta$.
Then:
the magnitude of $\mathbf v$ is unchanged in the new coordinate system. | We offer three equivalent statements:
By definition, rotation of the coordinate system affects the coordinates and not the vector.
{{tidy}}
Rotation is a rigid transformation. It does not change side lengths or angles.
The equations of rotation of coordinates are linear transformations.
{{qed}} | Let $\mathbf v$ be an arbitrary [[Definition:Vector Quantity|vector]] in the [[Definition:Cartesian Coordinate System|Cartesian plane]] $\CC$.
Let the [[Definition:Cartesian Coordinate System|coordinate system]] then be [[Definition:Rotation (Geometry)|rotated]] in the [[Definition:Anticlockwise|anticlockwise directio... | We offer three equivalent statements:
By definition, [[Definition:Rotation (Geometry)|rotation]] of the [[Definition:Cartesian Coordinate System|coordinate system]] affects the [[Definition:Cartesian Coordinate System|coordinates]] and not the [[Definition:Vector Quantity|vector]].
{{tidy}}
[[Definition:Rotation (Geom... | Vector Magnitude is Invariant Under Rotation/Proof 2 | https://proofwiki.org/wiki/Vector_Magnitude_is_Invariant_Under_Rotation | https://proofwiki.org/wiki/Vector_Magnitude_is_Invariant_Under_Rotation/Proof_2 | [
"Vector Magnitude is Invariant Under Rotation",
"Geometric Rotations",
"Vector Length"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Rotation (Geometry)",
"Definition:Anticlockwise",
"Definition:Angle",
"Definition:Magnitude",
"Definition:Cartesian Coordinate System"
] | [
"Definition:Rotation (Geometry)",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Vector Quantity",
"Definition:Rotation (Geometry)",
"Rigid Transformation",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Angle",
"Equation... |
proofwiki-21426 | Vector Sum of Rotated Triangle is Zero | Let $T = \triangle ABC$ be embedded in the Cartesian plane $\CC$.
Let the sides of $T$ be directed line segments, that is, vectors.
Let $\CC$ be rotated anticlockwise about the origin by an angle $\theta$.
Let the triangle $T$, as defined in the rotated coordinates, be T'.
Let $\mathbf{0}$ be the zero vector.
Let $+$ b... | Let $\mathbf{AB}$ have the same magnitude as $AB$ and direction from $A$ to $B$.
Let similar definitions hold for $\mathbf{BC}$ and $\mathbf{CA}$.
By definition of vectors:
:$\mathbf{AB} = \paren { B_x - A_x, B_y - A_y }$
Let $\mathbf{AB}$ in the rotated coordinates be designated $\mathbf{AB}'$.
By definition of vecto... | Let $T = \triangle ABC$ be embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$.
Let the [[Definition:Side of Polygon|sides]] of $T$ be [[Definition:Directed Line Segment|directed line segments]], that is, [[Definition:Vector Quantity|vectors]].
Let $\CC$ be [[Definition:Plane Rotation|rotated]] [[Def... | Let $\mathbf{AB}$ have the same [[Definition:Magnitude|magnitude]] as $AB$ and [[Definition:Direction|direction]] from $A$ to $B$.
Let similar definitions hold for $\mathbf{BC}$ and $\mathbf{CA}$.
By definition of [[Definition:Vector Quantity|vectors]]:
:$\mathbf{AB} = \paren { B_x - A_x, B_y - A_y }$
Let $\mathbf{... | Vector Sum of Rotated Triangle is Zero | https://proofwiki.org/wiki/Vector_Sum_of_Rotated_Triangle_is_Zero | https://proofwiki.org/wiki/Vector_Sum_of_Rotated_Triangle_is_Zero | [
"Vectors",
"Triangles"
] | [
"Definition:Cartesian Plane",
"Definition:Polygon/Side",
"Definition:Directed Line Segment",
"Definition:Vector Quantity",
"Definition:Rotation (Geometry)/Plane",
"Definition:Anticlockwise",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Triangle",
"Definition:Rotation (Geo... | [
"Definition:Magnitude",
"Definition:Direction",
"Definition:Vector Quantity",
"Definition:Rotation (Geometry)/Plane",
"Definition:Cartesian Coordinate System",
"Definition:Vector Quantity",
"Equations defining Plane Rotation",
"Vector Magnitude is Invariant Under Rotation",
"Vector Magnitude is Inva... |
proofwiki-21427 | Combination Theorem for Bounded Real-Valued Functions/Sum Rule | :$f + g$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
Let $M = M_f + M_g$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{f + g} } s}
|... | :$f + g$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
Let $M = M_f + M_g$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
... | Combination Theorem for Bounded Real-Valued Functions/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Sum_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Triangle Inequality/Real Numbers",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21428 | Combination Theorem for Bounded Real-Valued Functions/Negation Rule | :$-f$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M$
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{-f} } s}
| r = \size{-\map f s}
| c = {{Defof|Absolute Value of Real-Valued Function}}
}}
{{eqn | r = \size{\m... | :$-f$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M$
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{-f} } s}
| r = \size{-\map f s}
| c = {{Defof|Absolute Value of ... | Combination Theorem for Bounded Real-Valued Functions/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Negation_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Absolute Value of Negative",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21429 | Combination Theorem for Bounded Real-Valued Functions/Product Rule | :$f g$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
Let $M = M_f M_g$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{f g} } s}
| r =... | :$f g$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
Let $M = M_f M_g$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
... | Combination Theorem for Bounded Real-Valued Functions/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Product_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Absolute Value Function is Completely Multiplicative",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21430 | Combination Theorem for Bounded Real-Valued Functions/Absolute Value Rule | :$\size f$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M$
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \bigsize{\map {\size f} s}
| r = \bigsize{\size{\map f s} }
| c = Definition of $\size f$
}}
{{eqn | r = \size{\map f s}
| ... | :$\size f$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M$
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \bigsize{\map {\size f} s}
| r = \bigsize{\size{\map f s} }
| c = Definition of $\s... | Combination Theorem for Bounded Real-Valued Functions/Absolute Value Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Absolute_Value_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Absolute_Value_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21431 | Combination Theorem for Bounded Real-Valued Functions/Maximum Rule | :$f \vee g$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
From Negative of Absolute Value:
:$\forall s \in S : \map f s \le \size{\map f s}$
and
:$\forall s \in S : \map g s \le \s... | :$f \vee g$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
From [[Negative of Absolute Value]]:
:$\forall s \in S : \map f s \le \size{... | Combination Theorem for Bounded Real-Valued Functions/Maximum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Maximum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Maximum_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Negative of Absolute Value",
"Max Operation Preserves Total Ordering",
"Max Operation Preserves Total Ordering",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21432 | Combination Theorem for Bounded Real-Valued Functions/Minimum Rule | :$f \wedge g$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
From Negative of Absolute Value:
:$\forall s \in S : \map f s \le \size{\map f s}$
and
:$\forall s \in S : \map g s \le \s... | :$f \wedge g$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
and
:$\exists M_g \in \R_{\ge 0} : \forall s \in S : \size{\map g s} \le M_g$
From [[Negative of Absolute Value]]:
:$\forall s \in S : \map f s \le \size{... | Combination Theorem for Bounded Real-Valued Functions/Minimum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Minimum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Minimum_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Negative of Absolute Value",
"Min Operation Preserves Total Ordering",
"Min Operation Preserves Total Ordering",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21433 | Combination Theorem for Continuous Real-Valued Functions/Sum Rule | :$f + g$ is a coninuous real-valued function | Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Sum Rule for Continuous Mappings into Normed Division Ring
{{qed}}
Category:Combination Theorem for Continuous Real-Valued Functions
gw5n2d3n1xgh3w7j5xzw1r2uopx5130 | :$f + g$ is a [[Definition:Continuous Real-Valued Function|coninuous real-valued function]] | Follows from:
:[[Real Numbers form Valued Field]]
:By definition a [[Definition:Valued Field|valued field]] is a [[Definition:Normed Division Ring|normed division ring]]
:[[Sum Rule for Continuous Mappings into Normed Division Ring]]
{{qed}}
[[Category:Combination Theorem for Continuous Real-Valued Functions]]
gw5n2d3... | Combination Theorem for Continuous Real-Valued Functions/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Sum_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Real Numbers form Valued Field",
"Definition:Valued Field",
"Definition:Normed Division Ring",
"Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule",
"Category:Combination Theorem for Continuous Real-Valued Functions"
] |
proofwiki-21434 | Combination Theorem for Continuous Real-Valued Functions/Negation Rule | :$-f$ is a continuous real-valued function | Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Negation Rule for Continuous Mappings into Normed Division Ring
{{qed}}
Category:Combination Theorem for Continuous Real-Valued Functions
61huilxlqhgymhb81wpyfoi5z301sc2 | :$-f$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
:[[Real Numbers form Valued Field]]
:By definition a [[Definition:Valued Field|valued field]] is a [[Definition:Normed Division Ring|normed division ring]]
:[[Negation Rule for Continuous Mappings into Normed Division Ring]]
{{qed}}
[[Category:Combination Theorem for Continuous Real-Valued Functions]]
6... | Combination Theorem for Continuous Real-Valued Functions/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Negation_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Real Numbers form Valued Field",
"Definition:Valued Field",
"Definition:Normed Division Ring",
"Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule",
"Category:Combination Theorem for Continuous Real-Valued Functions"
] |
proofwiki-21435 | Combination Theorem for Continuous Real-Valued Functions/Product Rule | :$f g$ is a continuous real-valued function | Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Product Rule for Continuous Mappings into Normed Division Ring
{{qed}}
Category:Combination Theorem for Continuous Real-Valued Functions
ctgqh8g2eo5tsr62ctxwcwg4nizg8y4 | :$f g$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
:[[Real Numbers form Valued Field]]
:By definition a [[Definition:Valued Field|valued field]] is a [[Definition:Normed Division Ring|normed division ring]]
:[[Product Rule for Continuous Mappings into Normed Division Ring]]
{{qed}}
[[Category:Combination Theorem for Continuous Real-Valued Functions]]
ct... | Combination Theorem for Continuous Real-Valued Functions/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Product_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Real Numbers form Valued Field",
"Definition:Valued Field",
"Definition:Normed Division Ring",
"Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule",
"Category:Combination Theorem for Continuous Real-Valued Functions"
] |
proofwiki-21436 | Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule | :$\size f$ is a continuous real-valued function | From Absolute Value of Function is Composite with Absolute Value Function:
:$\size{f} = \size{\,\cdot\,} \circ f$
where:
:$\size{\,\cdot\,}$ denotes the absolute value function $\size{\,\cdot\,} : \R \to \R$
:$\size{\,\cdot\,} \circ f$ denotes the composite mapping of $\size{\,\cdot\,}$ with $f$
From Absolute Value Fun... | :$\size f$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | From [[Absolute Value of Function is Composite with Absolute Value Function]]:
:$\size{f} = \size{\,\cdot\,} \circ f$
where:
:$\size{\,\cdot\,}$ denotes the [[Definition:Absolute Value|absolute value function]] $\size{\,\cdot\,} : \R \to \R$
:$\size{\,\cdot\,} \circ f$ denotes the [[Definition:Composite Mapping|composi... | Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Absolute_Value_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Absolute_Value_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Absolute Value of Function is Composite with Absolute Value Function",
"Definition:Absolute Value",
"Definition:Composition of Mappings",
"Absolute Value Function is Continuous",
"Definition:Continuous",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)",
"Ca... |
proofwiki-21437 | Combination Theorem for Continuous Real-Valued Functions/Maximum Rule | :$f \vee g$ is a continuous real-valued function | From Characterization of Pointwise Maximum of Real-Valued Functions:
::$f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
We have:
{{begin-eqn}}
{{eqn | o =
| r = f, g \text{ are continuous real-valued functions}
}}
{{eqn | o = \leadsto
| r = f -g \text{ is a continuous real-valued function}
| c ... | :$f \vee g$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | From [[Characterization of Pointwise Maximum of Real-Valued Functions]]:
::$f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
We have:
{{begin-eqn}}
{{eqn | o =
| r = f, g \text{ are continuous real-valued functions}
}}
{{eqn | o = \leadsto
| r = f -g \text{ is a continuous real-valued function}
... | Combination Theorem for Continuous Real-Valued Functions/Maximum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Maximum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Maximum_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Characterization of Pointwise Maximum of Real-Valued Functions",
"Combination Theorem for Continuous Real-Valued Functions/Difference Rule",
"Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule",
"Combination Theorem for Continuous Real-Valued Functions/Sum Rule",
"Combination Theo... |
proofwiki-21438 | Combination Theorem for Continuous Real-Valued Functions/Minimum Rule | :$f \wedge g$ is a continuous real-valued function | From Characterization of Pointwise Minimum of Real-Valued Functions:
::$f \vee g = \dfrac 1 2 \paren{f + g - \size{f - g}}$
We have:
{{begin-eqn}}
{{eqn | o =
| r = f, g \text{ are continuous real-valued functions}
}}
{{eqn | o = \leadsto
| r = f -g \text{ is a continuous real-valued function}
| c ... | :$f \wedge g$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | From [[Characterization of Pointwise Minimum of Real-Valued Functions]]:
::$f \vee g = \dfrac 1 2 \paren{f + g - \size{f - g}}$
We have:
{{begin-eqn}}
{{eqn | o =
| r = f, g \text{ are continuous real-valued functions}
}}
{{eqn | o = \leadsto
| r = f -g \text{ is a continuous real-valued function}
... | Combination Theorem for Continuous Real-Valued Functions/Minimum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Minimum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Minimum_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Characterization of Pointwise Minimum of Real-Valued Functions",
"Combination Theorem for Continuous Real-Valued Functions/Difference Rule",
"Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule",
"Combination Theorem for Continuous Real-Valued Functions/Difference Rule",
"Combinati... |
proofwiki-21439 | Combination Theorem for Bounded Continuous Real-Valued Functions/Sum Rule | :$f + g$ is a bounded coninuous real-valued function | Follows from:
* Sum Rule for Bounded Real-Valued Functions
* Sum Rule for Continuous Real-Valued Functions
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
1eui55ymtva40xaz7z7q4mzo739y4q8 | :$f + g$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|coninuous real-valued function]] | Follows from:
* [[Sum Rule for Bounded Real-Valued Functions]]
* [[Sum Rule for Continuous Real-Valued Functions]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
1eui55ymtva40xaz7z7q4mzo739y4q8 | Combination Theorem for Bounded Continuous Real-Valued Functions/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Sum_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Sum Rule",
"Combination Theorem for Continuous Real-Valued Functions/Sum Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21440 | Combination Theorem for Bounded Continuous Real-Valued Functions/Negation Rule | :$-f$ is a bounded continuous real-valued function | Follows from:
* Negation Rule for Bounded Real-Valued Function
* Negation Rule for Continuous Real-Valued Function
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
3ib54dvxzda1tpxkfrhqf23skc37upi | :$-f$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Negation Rule for Bounded Real-Valued Function]]
* [[Negation Rule for Continuous Real-Valued Function]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
3ib54dvxzda1tpxkfrhqf23skc37upi | Combination Theorem for Bounded Continuous Real-Valued Functions/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Negation_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Negation Rule",
"Combination Theorem for Continuous Real-Valued Functions/Negation Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21441 | Combination Theorem for Bounded Continuous Real-Valued Functions/Product Rule | :$f g$ is a bounded continuous real-valued function | Follows from:
* Product Rule for Bounded Real-Valued Functions
* Product Rule for Continuous Real-Valued Functions
{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
gyep7v1ibm7q32uzdccowdwok87bjbr | :$f g$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Product Rule for Bounded Real-Valued Functions]]
* [[Product Rule for Continuous Real-Valued Functions]]
{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
gyep7v1ibm7q32uzdccowdwok87bjbr | Combination Theorem for Bounded Continuous Real-Valued Functions/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Product_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Product Rule",
"Combination Theorem for Continuous Real-Valued Functions/Product Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21442 | Combination Theorem for Bounded Continuous Real-Valued Functions/Absolute Value Rule | :$\size f$ is a bounded continuous real-valued function | Follows from:
* Absolute Value Rule for Bounded Real-Valued Function
* Absolute Value Rule for Continuous Real-Valued Function
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
gz0dda15em4azdr34ejal629bvabl5y | :$\size f$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Absolute Value Rule for Bounded Real-Valued Function]]
* [[Absolute Value Rule for Continuous Real-Valued Function]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
gz0dda15em4azdr34ejal629bvabl5y | Combination Theorem for Bounded Continuous Real-Valued Functions/Absolute Value Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Absolute_Value_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Absolute_Value_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Absolute Value Rule",
"Combination Theorem for Continuous Real-Valued Functions/Absolute Value Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21443 | Combination Theorem for Bounded Continuous Real-Valued Functions/Maximum Rule | :$f \vee g$ is a bounded continuous real-valued function | Follows from:
* Maximum Rule for Bounded Real-Valued Functions
* Maximum Rule for Continuous Real-Valued Functions
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
5dov7q0xan9kj94xfywaaqvmih6wi95 | :$f \vee g$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Maximum Rule for Bounded Real-Valued Functions]]
* [[Maximum Rule for Continuous Real-Valued Functions]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
5dov7q0xan9kj94xfywaaqvmih6wi95 | Combination Theorem for Bounded Continuous Real-Valued Functions/Maximum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Maximum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Maximum_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Maximum Rule",
"Combination Theorem for Continuous Real-Valued Functions/Maximum Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21444 | Combination Theorem for Bounded Continuous Real-Valued Functions/Minimum Rule | :$f \wedge g$ is a bounded continuous real-valued function | Follows from:
* Minimum Rule for Bounded Real-Valued Functions
* Minimum Rule for Continuous Real-Valued Functions
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
is078cyx8q5r0r920qp4l30v95d3m1r | :$f \wedge g$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Minimum Rule for Bounded Real-Valued Functions]]
* [[Minimum Rule for Continuous Real-Valued Functions]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
is078cyx8q5r0r920qp4l30v95d3m1r | Combination Theorem for Bounded Continuous Real-Valued Functions/Minimum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Minimum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Minimum_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Minimum Rule",
"Combination Theorem for Continuous Real-Valued Functions/Minimum Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21445 | Max Operation Preserves Total Ordering | Let $\struct {S, \preceq}$ be a totally ordered set.
Let $a, b, c, d \in S$:
:$a \preceq b, c \preceq d$
Then:
:$\max \set{a, c} \preceq \max \set{b, d}$
where $\max$ denotes the max operation on $\struct {S, \preceq}$. | From Max Operation Equals an Operand, either:
:$\max \set{a, c} = a$
or
:$\max \set{a, c} = c$
{{WLOG}}, suppose:
:$\max \set{a, c} = a$
We have:
{{begin-eqn}}
{{eqn | l = \max \set{a, c}
| r = a
| c = {{Hypothesis}}
}}
{{eqn | o = \preceq
| r = b
| c = {{Hypothesis}}
}}
{{eqn | o = \preceq
... | Let $\struct {S, \preceq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $a, b, c, d \in S$:
:$a \preceq b, c \preceq d$
Then:
:$\max \set{a, c} \preceq \max \set{b, d}$
where $\max$ denotes the [[Definition:Max Operation|max operation]] on $\struct {S, \preceq}$. | From [[Max Operation Equals an Operand]], either:
:$\max \set{a, c} = a$
or
:$\max \set{a, c} = c$
{{WLOG}}, suppose:
:$\max \set{a, c} = a$
We have:
{{begin-eqn}}
{{eqn | l = \max \set{a, c}
| r = a
| c = {{Hypothesis}}
}}
{{eqn | o = \preceq
| r = b
| c = {{Hypothesis}}
}}
{{eqn | o = \pre... | Max Operation Preserves Total Ordering | https://proofwiki.org/wiki/Max_Operation_Preserves_Total_Ordering | https://proofwiki.org/wiki/Max_Operation_Preserves_Total_Ordering | [
"Max Operation"
] | [
"Definition:Totally Ordered Set",
"Definition:Max Operation"
] | [
"Max Operation Equals an Operand",
"Max Operation Yields Supremum of Parameters",
"Definition:Transitive Relation",
"Definition:Ordering",
"Category:Max Operation"
] |
proofwiki-21446 | Min Operation Preserves Total Ordering | Let $\struct {S, \preceq}$ be a totally ordered set.
Let $a, b, c, d \in S$:
:$a \preceq b, c \preceq d$
Then:
:$\min \set{a, c} \preceq \min \set{b, d}$
where $\min$ denotes the min operation on $\struct {S, \preceq}$. | From Min Operation Equals an Operand, either:
:$\min \set{b, d} = b$
or
:$\min \set{b, d} = d$
{{WLOG}}, suppose:
:$\min \set{b, d} = b$
We have:
{{begin-eqn}}
{{eqn | l = \min \set{a, c}
| o = \preceq
| r = a
| c = Min Operation Yields Infimum of Parameters and {{Defof|Infimum}}
}}
{{eqn | o = \prece... | Let $\struct {S, \preceq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $a, b, c, d \in S$:
:$a \preceq b, c \preceq d$
Then:
:$\min \set{a, c} \preceq \min \set{b, d}$
where $\min$ denotes the [[Definition:Min Operation|min operation]] on $\struct {S, \preceq}$. | From [[Min Operation Equals an Operand]], either:
:$\min \set{b, d} = b$
or
:$\min \set{b, d} = d$
{{WLOG}}, suppose:
:$\min \set{b, d} = b$
We have:
{{begin-eqn}}
{{eqn | l = \min \set{a, c}
| o = \preceq
| r = a
| c = [[Min Operation Yields Infimum of Parameters]] and {{Defof|Infimum}}
}}
{{eqn ... | Min Operation Preserves Total Ordering | https://proofwiki.org/wiki/Min_Operation_Preserves_Total_Ordering | https://proofwiki.org/wiki/Min_Operation_Preserves_Total_Ordering | [
"Min Operation"
] | [
"Definition:Totally Ordered Set",
"Definition:Min Operation"
] | [
"Min Operation Equals an Operand",
"Min Operation Yields Infimum of Parameters",
"Definition:Transitive Relation",
"Definition:Ordering",
"Category:Min Operation"
] |
proofwiki-21447 | Characterization of Pointwise Maximum of Real-Valued Functions | Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be real-valued functions.
Let $f \vee g$ denote the pointwise maximum of $f$ and $g$, that is, $f \vee g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f \vee g} } s = \max \set{\map f s, \map g s}$
Then:
:$f \vee g = \dfrac 1 2 ... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren {f \vee g} } s
| r = \max \set {\map f x, \map g x}
| c = {{Defof|Pointwise Maximum of Real-Valued Functions}}
}}
{{eqn | r = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }
| c = Max is Half of Sum Plus... | Let $S$ be a [[Definition:Set|set]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $f, g :S \to \R$ be [[Definition:Real-Valued Function|real-valued functions]].
Let $f \vee g$ denote the [[Definition:Pointwise Maximum of Real-Valued Functions|pointwise maximum]] of $f$ and $g$, that is, ... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren {f \vee g} } s
| r = \max \set {\map f x, \map g x}
| c = {{Defof|Pointwise Maximum of Real-Valued Functions}}
}}
{{eqn | r = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }
| c = [[Max is Half of Sum Pl... | Characterization of Pointwise Maximum of Real-Valued Functions | https://proofwiki.org/wiki/Characterization_of_Pointwise_Maximum_of_Real-Valued_Functions | https://proofwiki.org/wiki/Characterization_of_Pointwise_Maximum_of_Real-Valued_Functions | [
"Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Real Number/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Pointwise Maximum of Mappings/Real-Valued Functions",
"Definition:Mapping",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Pointwise Difference of Real-Valued Functions",
... | [
"Max is Half of Sum Plus Absolute Difference",
"Equality of Mappings",
"Category:Real-Valued Functions"
] |
proofwiki-21448 | Orthocenter and Incenter Coincide if Triangle is Equilateral | Let $\triangle ABC$ be an equilateral triangle.
Let $G$ be the orthocenter of $\triangle ABC$.
Then:
$G$ is the incenter of $\triangle ABC$. | Draw the circumcircle of $\triangle ABC$ through points $A, B,$ and $C$.
300px
Given: $G$ is the orthocenter of $\triangle ABC$.
By Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral:
:$G$ is also the centroid and circumcenter of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = AB
| r = AC = C... | Let $\triangle ABC$ be an [[Definition:Equilateral Triangle|equilateral triangle]].
Let $G$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
Then:
$G$ is the [[Definition:Incenter of Triangle|incenter]] of $\triangle ABC$. | Draw the [[Definition:Circumcircle|circumcircle]] of $\triangle ABC$ through points $A, B,$ and $C$.
[[File:CircumscribedEquilateral.png|300px]]
Given: $G$ is the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
By [[Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral]]:
:$G$ is al... | Orthocenter and Incenter Coincide if Triangle is Equilateral | https://proofwiki.org/wiki/Orthocenter_and_Incenter_Coincide_if_Triangle_is_Equilateral | https://proofwiki.org/wiki/Orthocenter_and_Incenter_Coincide_if_Triangle_is_Equilateral | [
"Equilateral Triangles",
"Orthocenters of Triangles",
"Incircles of Triangles"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Orthocenter",
"Definition:Incircle of Triangle/Incenter"
] | [
"Definition:Circumcircle",
"File:CircumscribedEquilateral.png",
"Definition:Orthocenter",
"Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral",
"Medians of Triangle Meet at Centroid",
"Triangle Right-Angle-Hypotenuse-Side Congruence",
"Triangle Side-Side-Side Congruence",
"Def... |
proofwiki-21449 | Characterization of Pointwise Minimum of Real-Valued Functions | Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be real-valued functions.
Let $f \wedge g$ denote the pointwise maximum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:
:$\forall s \in S : \map {\paren {f \wedge g} } s = \min \set {\map f s, \map g s}$
Then:
:$f \wedge g = \... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren {f \wedge g} } s
| r = \min \set {\map f s, \map g s}
| c = {{Defof|Pointwise Minimum of Real-Valued Functions}}
}}
{{eqn | r = \dfrac 1 2 \paren {\map f s + \map g x - \size {\map f x - \map g x} }
| c = Min is Half of Sum Le... | Let $S$ be a [[Definition:Set|set]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $f, g :S \to \R$ be [[Definition:Real-Valued Function|real-valued functions]].
Let $f \wedge g$ denote the [[Definition:Pointwise Minimum of Real-Valued Functions|pointwise maximum]] of $f$ and $g$, that is... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren {f \wedge g} } s
| r = \min \set {\map f s, \map g s}
| c = {{Defof|Pointwise Minimum of Real-Valued Functions}}
}}
{{eqn | r = \dfrac 1 2 \paren {\map f s + \map g x - \size {\map f x - \map g x} }
| c = [[Min is Half of Sum ... | Characterization of Pointwise Minimum of Real-Valued Functions | https://proofwiki.org/wiki/Characterization_of_Pointwise_Minimum_of_Real-Valued_Functions | https://proofwiki.org/wiki/Characterization_of_Pointwise_Minimum_of_Real-Valued_Functions | [
"Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Real Number/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Pointwise Minimum of Mappings/Real-Valued Functions",
"Definition:Mapping",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Pointwise Difference of Real-Valued Functions",
... | [
"Min is Half of Sum Less Absolute Difference",
"Equality of Mappings",
"Category:Real-Valued Functions"
] |
proofwiki-21450 | Combination Theorem for Continuous Real-Valued Functions/Difference Rule | :$f - g$ is a coninuous real-valued function | From Pointwise Difference is Pointwise Addition with Negation:
:$f - g = f + \paren{-g}$
where:
:$-g$ denotes the pointwise negation of $g$
:$f + \paren{-g}$ denotes the pointwise addition of $f$ and $-g$
From Negation Rule for Continuous Real-Valued Function:
:$-g$ is a contiuous real-valued function
From Sum Rule for... | :$f - g$ is a [[Definition:Continuous Real-Valued Function|coninuous real-valued function]] | From [[Pointwise Difference is Pointwise Addition with Negation]]:
:$f - g = f + \paren{-g}$
where:
:$-g$ denotes the [[Definition:Pointwise Negation of Real-Valued Function|pointwise negation]] of $g$
:$f + \paren{-g}$ denotes the [[Definition:Pointwise Addition of Real-Valued Functions|pointwise addition]] of $f$ and... | Combination Theorem for Continuous Real-Valued Functions/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Difference_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Pointwise Difference is Pointwise Addition with Negation",
"Definition:Pointwise Negation of Real-Valued Function",
"Definition:Pointwise Addition of Real-Valued Functions",
"Combination Theorem for Continuous Real-Valued Functions/Negation Rule",
"Definition:Continuous Real-Valued Vector Function",
"Com... |
proofwiki-21451 | Combination Theorem for Continuous Real-Valued Functions/Multiple Rule | :$\lambda f$ is a continuous real-valued function | Follows from:
:Real Numbers form Valued Field
:By definition a valued field is a normed division ring
:Multiple Rule for Continuous Mappings into Normed Division Ring
{{qed}}
Category:Combination Theorem for Continuous Real-Valued Functions
jjr6i2ji3qfpjbsqavgzee86ruusar1 | :$\lambda f$ is a [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
:[[Real Numbers form Valued Field]]
:By definition a [[Definition:Valued Field|valued field]] is a [[Definition:Normed Division Ring|normed division ring]]
:[[Multiple Rule for Continuous Mappings into Normed Division Ring]]
{{qed}}
[[Category:Combination Theorem for Continuous Real-Valued Functions]]
j... | Combination Theorem for Continuous Real-Valued Functions/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Real-Valued_Functions/Multiple_Rule | [
"Combination Theorem for Continuous Real-Valued Functions"
] | [
"Definition:Continuous Real-Valued Vector Function"
] | [
"Real Numbers form Valued Field",
"Definition:Valued Field",
"Definition:Normed Division Ring",
"Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule",
"Category:Combination Theorem for Continuous Real-Valued Functions"
] |
proofwiki-21452 | Real Numbers form Valued Field | The set of real numbers $\R$ forms a valued field under addition, multiplication and absolute value: $\struct {\R, +, \times, \size {\,\cdot\,} }$. | From Real Numbers form Field, we have that $\struct {\R, +, \times}$ forms a field.
From Absolute Value is Norm, we have that $\size {\size {\,\cdot\,} }$ is a norm on $\struct {\R, +, \times}$.
Hence $\struct {\R, +, \times, \size {\,\cdot\,} }$ is a valued field by definition.
{{qed}} | The [[Definition:Real Number|set of real numbers]] $\R$ forms a [[Definition:Valued Field|valued field]] under [[Definition:Real Addition|addition]], [[Definition:Real Multiplication|multiplication]] and [[Definition:Absolute Value|absolute value]]: $\struct {\R, +, \times, \size {\,\cdot\,} }$. | From [[Real Numbers form Field]], we have that $\struct {\R, +, \times}$ forms a [[Definition:Field (Abstract Algebra)|field]].
From [[Absolute Value is Norm]], we have that $\size {\size {\,\cdot\,} }$ is a [[Definition:Norm on Division Ring|norm]] on $\struct {\R, +, \times}$.
Hence $\struct {\R, +, \times, \size {... | Real Numbers form Valued Field | https://proofwiki.org/wiki/Real_Numbers_form_Valued_Field | https://proofwiki.org/wiki/Real_Numbers_form_Valued_Field | [
"Examples of Fields",
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Valued Field",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers",
"Definition:Absolute Value"
] | [
"Real Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Absolute Value is Norm",
"Definition:Norm/Division Ring",
"Definition:Valued Field"
] |
proofwiki-21453 | Absolute Value Function is Continuous | Let $f$ be the real function defined as:
:$\forall x \in \R: \map f x = \size x$
where $\size x$ denotes the absolute value of $x$.
Then $f$ is a continuous real function. | Let $a \in \R$.
Let $\epsilon \in \R_{\mathop > 0}$.
Let $\delta \le \epsilon$.
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in \R : \size{x - a} < \delta
| l = \size{\map f x - \map f a}
| r = \bigsize {\size x - \size a}
| c = Definition of $f$
}}
{{eqn | o = \le
| r = \size {x - a}
| c... | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map f x = \size x$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$.
Then $f$ is a [[Definition:Everywhere Continuous Real Function|continuous real function]]. | Let $a \in \R$.
Let $\epsilon \in \R_{\mathop > 0}$.
Let $\delta \le \epsilon$.
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in \R : \size{x - a} < \delta
| l = \size{\map f x - \map f a}
| r = \bigsize {\size x - \size a}
| c = Definition of $f$
}}
{{eqn | o = \le
| r = \size {x - a}
... | Absolute Value Function is Continuous | https://proofwiki.org/wiki/Absolute_Value_Function_is_Continuous | https://proofwiki.org/wiki/Absolute_Value_Function_is_Continuous | [
"Absolute Value Function",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Absolute Value",
"Definition:Continuous Real Function/Everywhere"
] | [
"Reverse Triangle Inequality/Real and Complex Fields",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Everywhere",
"Category:Absolute Value Function",
"Category:Continuous Real Functions"
] |
proofwiki-21454 | Pointwise Difference is Pointwise Addition with Negation | Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be real-valued functions.
Let $f - g$ denote the pointwise difference of $f$ and $g$, that is, $f - g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f - g} } s = \map f s - \map g s$
Then:
:$f - g = f + \paren{-g}$
where:
:$-g$ d... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren{f - g} } s
| r = \map f s - \map g s
| c = {{Defof|Pointwise Difference of Real-Valued Functions}}
}}
{{eqn | r = \map f s + \paren{- \map g s}
}}
{{eqn | r = \map f s + \map {\paren{-g} } s
| c = {{Defof|Pointwise Negation of... | Let $S$ be a [[Definition:Set|set]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $f, g :S \to \R$ be [[Definition:Real-Valued Function|real-valued functions]].
Let $f - g$ denote the [[Definition:Pointwise Difference of Real-Valued Functions|pointwise difference]] of $f$ and $g$, that i... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\paren{f - g} } s
| r = \map f s - \map g s
| c = {{Defof|Pointwise Difference of Real-Valued Functions}}
}}
{{eqn | r = \map f s + \paren{- \map g s}
}}
{{eqn | r = \map f s + \map {\paren{-g} } s
| c = {{Defof|Pointwise Negation of... | Pointwise Difference is Pointwise Addition with Negation | https://proofwiki.org/wiki/Pointwise_Difference_is_Pointwise_Addition_with_Negation | https://proofwiki.org/wiki/Pointwise_Difference_is_Pointwise_Addition_with_Negation | [
"Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Real Number/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Pointwise Difference of Real-Valued Functions",
"Definition:Mapping",
"Definition:Pointwise Negation of Real-Valued Function",
"Definition:Pointwise Addition of Real-Valued Functions"
] | [
"Equality of Mappings",
"Category:Real-Valued Functions"
] |
proofwiki-21455 | Absolute Value of Function is Composite with Absolute Value Function | Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f: S \to \R$ be real-valued function.
Let $\size f$ denote the absolute value of $f$, that is, $\size f$ is the mapping defined by:
:$\forall s \in S : \map {\size f} s = \size{\map f s}$
Then:
:$\size f = \size{\,\cdot\,} \circ f$
where:
:$\size{\,\cdot\,} :... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\size f } s
| r = \size{\map f s}
| c = {{Defof|Absolute Value of Real-Valued Function}}
}}
{{eqn | r = \map {\paren{\size{\,\cdot\,} \circ f} } s
| c = {{Defof|Composite Mapping}}
}}
{{end-eqn}}
By definition of equality of mappings... | Let $S$ be a [[Definition:Set|set]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $f: S \to \R$ be [[Definition:Real-Valued Function|real-valued function]].
Let $\size f$ denote the [[Definition:Absolute Value of Real-Valued Function|absolute value]] of $f$, that is, $\size f$ is the [[D... | We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \map {\size f } s
| r = \size{\map f s}
| c = {{Defof|Absolute Value of Real-Valued Function}}
}}
{{eqn | r = \map {\paren{\size{\,\cdot\,} \circ f} } s
| c = {{Defof|Composite Mapping}}
}}
{{end-eqn}}
By definition of [[Definition:Equal... | Absolute Value of Function is Composite with Absolute Value Function | https://proofwiki.org/wiki/Absolute_Value_of_Function_is_Composite_with_Absolute_Value_Function | https://proofwiki.org/wiki/Absolute_Value_of_Function_is_Composite_with_Absolute_Value_Function | [
"Real-Valued Functions"
] | [
"Definition:Set",
"Definition:Real Number/Real Number Line",
"Definition:Real-Valued Function",
"Definition:Absolute Value of Mapping/Real-Valued Function",
"Definition:Mapping",
"Definition:Absolute Value",
"Definition:Composition of Mappings"
] | [
"Equality of Mappings",
"Category:Real-Valued Functions"
] |
proofwiki-21456 | Combination Theorem for Bounded Real-Valued Functions/Difference Rule | :$f - g$ is a bounded real-valued function | From Pointwise Difference is Pointwise Addition with Negation:
:$f - g = f + \paren{-g}$
where:
:$-g$ denotes the pointwise negation of $g$
:$f + \paren{-g}$ denotes the pointwise addition of $f$ and $-g$
From Negation Rule for Bounded Real-Valued Function:
:$-g$ is a bounded real-valued function
From Sum Rule for Boun... | :$f - g$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | From [[Pointwise Difference is Pointwise Addition with Negation]]:
:$f - g = f + \paren{-g}$
where:
:$-g$ denotes the [[Definition:Pointwise Negation of Real-Valued Function|pointwise negation]] of $g$
:$f + \paren{-g}$ denotes the [[Definition:Pointwise Addition of Real-Valued Functions|pointwise addition]] of $f$ and... | Combination Theorem for Bounded Real-Valued Functions/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Difference_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Pointwise Difference is Pointwise Addition with Negation",
"Definition:Pointwise Negation of Real-Valued Function",
"Definition:Pointwise Addition of Real-Valued Functions",
"Combination Theorem for Bounded Real-Valued Functions/Negation Rule",
"Definition:Bounded Mapping/Real-Valued",
"Combination Theor... |
proofwiki-21457 | Combination Theorem for Bounded Real-Valued Functions/Multiple Rule | :$\lambda f$ is a bounded real-valued function | By definition of bounded real-valued function
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
Let $M = \size{\lambda} M_f$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{\lambda f} } s}
| r = \size{\lambda \map f s}
| c = {{Defof|Pointwise Scalar... | :$\lambda f$ is a [[Definition:Bounded Real-Valued Function|bounded real-valued function]] | By definition of [[Definition:Bounded Real-Valued Function|bounded real-valued function]]
:$\exists M_f \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M_f$
Let $M = \size{\lambda} M_f$.
We have:
{{begin-eqn}}
{{eqn | q = \forall s \in S
| l = \size{\map {\paren{\lambda f} } s}
| r = \size{\lambd... | Combination Theorem for Bounded Real-Valued Functions/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Real-Valued_Functions/Multiple_Rule | [
"Combination Theorem for Bounded Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Absolute Value Function is Completely Multiplicative",
"Definition:Bounded Mapping/Real-Valued",
"Category:Combination Theorem for Bounded Real-Valued Functions"
] |
proofwiki-21458 | Inverse of a Line Through Circle Center is a Line | Let $C$ be a circle in the plane on center $O$.
Let the radius of $C$ be $r$.
Let $L$ be an arbitrary ray through $O$.
Let $P$ be an otherwise arbitrary point, but lying on $L$.
Let $T : X \to Y$ be an inversive transformation with $C$ as the inversion circle.
Then $O$ is the inversion center.
Then the image of $P$ un... | Let $P'$ be the image of $P$ under $T$, with $P \ne O$.
By the definition of inversion:
:$P'$ lies on $L$
We deal with $O$ separately. By the definition of ray:
:$O$ lies on $L$
{{EuclidSaid}}
:''{{:Definition:Euclid's Definitions - Book I/1 - Point}}''
A point has no length, it has no magnitude.
So we cannot compute ... | Let $C$ be a [[Definition:Circle|circle]] in the [[Definition:Plane|plane]] on [[Definition:Center of Circle|center]] $O$.
Let the [[Definition:Radius|radius]] of $C$ be $r$.
Let $L$ be an arbitrary [[Definition:Line|ray]] through $O$.
Let $P$ be an otherwise arbitrary [[Definition:Point|point]], but lying on $L$.
... | Let $P'$ be the [[Definition:Image of Element under Mapping|image]] of $P$ under $T$, with $P \ne O$.
By the definition of [[Definition:Inversive Transformation|inversion]]:
:$P'$ lies on $L$
We deal with $O$ separately. By the definition of [[Definition:Line|ray]]:
:$O$ lies on $L$
{{EuclidSaid}}
:''{{:Definition:... | Inverse of a Line Through Circle Center is a Line | https://proofwiki.org/wiki/Inverse_of_a_Line_Through_Circle_Center_is_a_Line | https://proofwiki.org/wiki/Inverse_of_a_Line_Through_Circle_Center_is_a_Line | [
"Inversive Geometry"
] | [
"Definition:Circle",
"Definition:Plane Surface",
"Definition:Circle/Center",
"Definition:Radius",
"Definition:Line",
"Definition:Point",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Inversive Transformation/Inversion Center",
"Definitio... | [
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Inversive Transformation",
"Definition:Line",
"Definition:Point",
"Definition:Linear Measure/Length",
"Definition:Magnitude",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Inversive Transformation",
"Definition:Image (Set Th... |
proofwiki-21459 | Compound Distribution of Poisson Distributed Bernoulli Trials has Poisson Distribution | Let $N$ be a discrete random variable with a Poisson distribution with expectation $\lambda$.
Let $X_1, X_2, \ldots, X_N$ be pairwise independent discrete random variables each with a Bernoulli distribution with parameter $P$.
Let $S_N : X_1 + X_2 + \cdots + X_N$ be the resulting compound distribution.
Then $S_N$ has a... | From the definition of Poisson distribution, we will show that:
:$\map \Pr {S_N = k} = \dfrac {\paren {\lambda p}^k e^{-\lambda p} } {k!}$
{{begin-eqn}}
{{eqn | l = \map \Pr {S_N = k}
| r = \sum_{n \mathop = 0}^\infty \condprob {S_N = k} {N = n} \map \Pr {N = n}
| c = Total Probability Theorem
}}
{{eqn | r ... | Let $N$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Poisson Distribution|Poisson distribution]] with [[Definition:Expectation|expectation]] $\lambda$.
Let $X_1, X_2, \ldots, X_N$ be [[Definition:Pairwise Independent Random Variables|pairwise independent]] [[Definition:Disc... | From the definition of [[Definition:Poisson Distribution|Poisson distribution]], we will show that:
:$\map \Pr {S_N = k} = \dfrac {\paren {\lambda p}^k e^{-\lambda p} } {k!}$
{{begin-eqn}}
{{eqn | l = \map \Pr {S_N = k}
| r = \sum_{n \mathop = 0}^\infty \condprob {S_N = k} {N = n} \map \Pr {N = n}
| c = [... | Compound Distribution of Poisson Distributed Bernoulli Trials has Poisson Distribution | https://proofwiki.org/wiki/Compound_Distribution_of_Poisson_Distributed_Bernoulli_Trials_has_Poisson_Distribution | https://proofwiki.org/wiki/Compound_Distribution_of_Poisson_Distributed_Bernoulli_Trials_has_Poisson_Distribution | [
"Compound Distributions",
"Poisson Distribution",
"Bernoulli Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Poisson Distribution",
"Definition:Expectation",
"Definition:Pairwise Independent Random Variables",
"Definition:Random Variable/Discrete",
"Definition:Bernoulli Distribution",
"Definition:Compound Distribution",
"Definition:Poisson Distribution",
"... | [
"Definition:Poisson Distribution",
"Total Probability Theorem",
"Definition:Support of Random Variable",
"Definition:Binomial Distribution",
"Binomial Experiment has Binomial Distribution",
"Translation of Index Variable of Summation/Infinite Series",
"Power Series Expansion for Exponential Function",
... |
proofwiki-21460 | Combination Theorem for Bounded Continuous Real-Valued Functions/Difference Rule | :$f - g$ is a bounded coninuous real-valued function | Follows from:
* Difference Rule for Bounded Real-Valued Functions
* Difference Rule for Continuous Real-Valued Functions
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
pjw75utujkv5upyi413ly1vowj1zogq | :$f - g$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|coninuous real-valued function]] | Follows from:
* [[Difference Rule for Bounded Real-Valued Functions]]
* [[Difference Rule for Continuous Real-Valued Functions]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
pjw75utujkv5upyi413ly1vowj1zogq | Combination Theorem for Bounded Continuous Real-Valued Functions/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Difference_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Difference Rule",
"Combination Theorem for Continuous Real-Valued Functions/Difference Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21461 | Combination Theorem for Bounded Continuous Real-Valued Functions/Multiple Rule | :$\lambda f$ is a bounded continuous real-valued function | Follows from:
* Multiple Rule for Bounded Real-Valued Function
* Multiple Rule for Continuous Real-Valued Function
{{qed}}
Category:Combination Theorem for Bounded Continuous Real-Valued Functions
gq7dnvedl3zmj7rwtz6s7efdhhtdqny | :$\lambda f$ is a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Continuous Real-Valued Function|continuous real-valued function]] | Follows from:
* [[Multiple Rule for Bounded Real-Valued Function]]
* [[Multiple Rule for Continuous Real-Valued Function]]
{{qed}}
[[Category:Combination Theorem for Bounded Continuous Real-Valued Functions]]
gq7dnvedl3zmj7rwtz6s7efdhhtdqny | Combination Theorem for Bounded Continuous Real-Valued Functions/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Bounded_Continuous_Real-Valued_Functions/Multiple_Rule | [
"Combination Theorem for Bounded Continuous Real-Valued Functions"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Continuous Real-Valued Vector Function"
] | [
"Combination Theorem for Bounded Real-Valued Functions/Multiple Rule",
"Combination Theorem for Continuous Real-Valued Functions/Multiple Rule",
"Category:Combination Theorem for Bounded Continuous Real-Valued Functions"
] |
proofwiki-21462 | Slope of Parabola at Point | Let $a$ be a constant.
Let $T$ be the parabola which is the locus of points $\tuple {x, y}$ satisfying $y = a x^2$.
The slope of the tangent to $y = a x^2$ at $x = c$ is $2 a c$. | By Derivative of Power of Function the derivative of $y = ax^2$ is:
:$\dfrac {\d y} {\d x} = 2 a x$
So the derivative of $y = a x^2$ at $x = c$ is $2 a c$.
The result follows from Slope of Tangent to Curve at Point equals Value of Derivative.
{{qed}}
Category:Parabolas
Category:Slope
n07nttxq0l26gdhy3rudjio4vj8g5a0 | Let $a$ be a [[Definition:Constant|constant]].
Let $T$ be the [[Definition:Parabola|parabola]] which is the [[Definition:Locus|locus]] of [[Definition:Point|points]] $\tuple {x, y}$ satisfying $y = a x^2$.
The [[Definition:Slope of Straight Line|slope]] of the [[Definition:Geometry|tangent]] to $y = a x^2$ at $x = c$... | By [[Derivative of Power of Function]] the [[Definition:Derivative|derivative]] of $y = ax^2$ is:
:$\dfrac {\d y} {\d x} = 2 a x$
So the [[Definition:Derivative|derivative]] of $y = a x^2$ at $x = c$ is $2 a c$.
The result follows from [[Slope of Tangent to Curve at Point equals Value of Derivative]].
{{qed}}
[[Cate... | Slope of Parabola at Point | https://proofwiki.org/wiki/Slope_of_Parabola_at_Point | https://proofwiki.org/wiki/Slope_of_Parabola_at_Point | [
"Parabolas",
"Slope"
] | [
"Definition:Constant",
"Definition:Parabola",
"Definition:Locus",
"Definition:Point",
"Definition:Slope/Straight Line",
"Definition:Geometry"
] | [
"Derivative of Power of Function",
"Definition:Derivative",
"Definition:Derivative",
"Slope of Tangent to Curve at Point equals Value of Derivative",
"Category:Parabolas",
"Category:Slope"
] |
proofwiki-21463 | Additive Inverse in Ring of Continuous Mappings | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with zero $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Let $f \in \map C {S, R}$.
Then:
:the additive inverse of $f$ is the pointwise negation $-f$ defined by:... | Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$
From Structure Induced by Ring Operations is Ring:
:$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise neg... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\struct {R, +, *, \tau_{_R} }$ be a [[Definition:Topological Ring|topological ring]] with [[Definition:Ring Zero|zero]] $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the [[Definition:Ring of Continuous Mappings|ring of continuo... | Let $\struct {R^S, +, *}$ be the [[Definition:Ring of Mappings|ring of mappings]] from $S$ to $R$.
From [[Ring of Continuous Mappings is Subring of All Mappings]]:
:$\struct{\map C {S, R}, +, *}$ is a [[Definition:Subring|subring]] of $\struct {R^S, +, *}$
From [[Structure Induced by Ring Operations is Ring]]:
:$\for... | Additive Inverse in Ring of Continuous Mappings | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Continuous_Mappings | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Continuous_Mappings | [
"Rings of Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Ring Zero",
"Definition:Ring of Continuous Mappings",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function"
] | [
"Definition:Ring of Mappings",
"Ring of Continuous Mappings is Subring of All Mappings",
"Definition:Subring",
"Structure Induced by Ring Operations is Ring",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function"
] |
proofwiki-21464 | Additive Inverse in Ring of Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Let $f \in \map C {S, \R}$.
Then:
:the additive inverse of $f$ is the pointwise negation $-f$ defined by:
::$\forall s \in S : \map {-f} s ... | By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Additive Inverse in Ring of Continuous Mappings:
:$\forall f \in \map C {S, \R} :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
::$\forall s \... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Let $f \in \map... | By definition of [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]]:
:$\struct {\map C {S, \R}, +, *}$ is the [[Definition:Ring of Continuous Mappings|ring of continuous mappings]] from $S$ to $\R$.
From [[Additive Inverse in Ring of Continuous Mappings]]:
:$\forall f \in ... | Additive Inverse in Ring of Continuous Real-Valued Functions | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Continuous_Real-Valued_Functions | [
"Rings of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Continuous Mappings",
"Additive Inverse in Ring of Continuous Mappings",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function"
] |
proofwiki-21465 | Additive Inverse in Ring of Bounded Continuous Real-Valued Functions | Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Let $f \in \map {C^*} {S, \R}$.
Then:
:the additive inverse of $f$ is the pointwise negation $-f$ defined by:
::$\forall s \in ... | Let $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a subring of $\struct{\map C {S, \R}, +, *}$
From Additive Inverse in Ring of Continuous Real-Va... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the [[Definition:Ring of Bounded Continuous Real-Valued Functions|ring of bounded continuous real-valued functions]] from $... | Let $\struct{\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions from $S$]].
From [[Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions]]:
:$\struct{\map {C^*} {S, \R}, +, *}$ is a [[Definition:Subring|subring]] of... | Additive Inverse in Ring of Bounded Continuous Real-Valued Functions | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Bounded_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Additive_Inverse_in_Ring_of_Bounded_Continuous_Real-Valued_Functions | [
"Rings of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Bounded Continuous Real-Valued Functions",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function"
] | [
"Definition:Ring of Continuous Real-Valued Functions",
"Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions",
"Definition:Subring",
"Additive Inverse in Ring of Continuous Real-Valued Functions",
"Definition:Additive Inverse",
"Definition:Pointwise Negation of Real-Valued F... |
proofwiki-21466 | Broken Chord Theorem | :300px
Let $A$ and $C$ be arbitrary points on a circle in the plane.
Let $M$ be a point on the circle with arc $AM = $ arc $MC$.
Let $B$ lie on the minor arc of $AM$.
Draw chords $AB$ and $BC$.
Find $D$ such that $MD \perp BC$.
Then:
:$AB + BD = DC$ | :300px
Let $E$ be a point such that $BD = DE$.
Given:
:arc $AM = $ arc $MC$
By Equal Arcs of Circles Subtended by Equal Straight Lines:
:$AM = MC$
By Angles on Equal Arcs are Equal:
:$\angle BAM = \angle MCB$
:$BM$ is shared
We have Ambiguous Case for Triangle Side-Side-Angle Congruence for these three triangles:
* $\t... | :[[File:Broken Chord.png|300px]]
Let $A$ and $C$ be arbitrary [[Definition:Point|points]] on a [[Definition:Circle|circle]] in the [[Definition:Cartesian Plane|plane]].
Let $M$ be a point on the circle with [[Definition:Arc of Circle|arc]] $AM = $ [[Definition:Arc of Circle|arc]] $MC$.
Let $B$ lie on the [[Definiti... | :[[File:Broken Chord 3.png|300px]]
Let $E$ be a [[Definition:Point|point]] such that $BD = DE$.
Given:
:[[Definition:Arc of Circle|arc]] $AM = $ [[Definition:Arc of Circle|arc]] $MC$
By [[Equal Arcs of Circles Subtended by Equal Straight Lines]]:
:$AM = MC$
By [[Angles on Equal Arcs are Equal]]:
:$\angle BAM = \ang... | Broken Chord Theorem/Proof 3 | https://proofwiki.org/wiki/Broken_Chord_Theorem | https://proofwiki.org/wiki/Broken_Chord_Theorem/Proof_3 | [
"Broken Chord Theorem",
"Circles",
"Named Theorems"
] | [
"File:Broken Chord.png",
"Definition:Point",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Minor Arc",
"Definition:Circle/Chord"
] | [
"File:Broken Chord 3.png",
"Definition:Point",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Equal Arcs of Circles Subtended by Equal Straight Lines",
"Angles on Equal Arcs are Equal",
"Ambiguous Case for Triangle Side-Side-Angle Congruence",
"Definition:Right Angle",
"External Angle of Triangl... |
proofwiki-21467 | Broken Chord Theorem | :300px
Let $A$ and $C$ be arbitrary points on a circle in the plane.
Let $M$ be a point on the circle with arc $AM = $ arc $MC$.
Let $B$ lie on the minor arc of $AM$.
Draw chords $AB$ and $BC$.
Find $D$ such that $MD \perp BC$.
Then:
:$AB + BD = DC$ | Find $E$ on $BC$ such that $BD = BE$.
400px
{{begin-eqn}}
{{eqn | l = BD
| r = ED
| c = {{hypothesis}}
}}
{{eqn | l = MD
| o = \perp
| r = BE
| c = {{hypothesis}}
}}
{{eqn | l = \triangle MBD
| o = \cong
| r = \triangle MED
| c = Triangle Side-Angle-Side Congruence
}}
{... | :[[File:Broken Chord.png|300px]]
Let $A$ and $C$ be arbitrary [[Definition:Point|points]] on a [[Definition:Circle|circle]] in the [[Definition:Cartesian Plane|plane]].
Let $M$ be a point on the circle with [[Definition:Arc of Circle|arc]] $AM = $ [[Definition:Arc of Circle|arc]] $MC$.
Let $B$ lie on the [[Definiti... | Find $E$ on $BC$ such that $BD = BE$.
[[File:Broken Chord 4.png|400px]]
{{begin-eqn}}
{{eqn | l = BD
| r = ED
| c = {{hypothesis}}
}}
{{eqn | l = MD
| o = \perp
| r = BE
| c = {{hypothesis}}
}}
{{eqn | l = \triangle MBD
| o = \cong
| r = \triangle MED
| c = [[Triangle... | Broken Chord Theorem/Proof 4 | https://proofwiki.org/wiki/Broken_Chord_Theorem | https://proofwiki.org/wiki/Broken_Chord_Theorem/Proof_4 | [
"Broken Chord Theorem",
"Circles",
"Named Theorems"
] | [
"File:Broken Chord.png",
"Definition:Point",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Minor Arc",
"Definition:Circle/Chord"
] | [
"File:Broken Chord 4.png",
"Triangle Side-Angle-Side Congruence",
"Definition:Congruence (Geometry)",
"Isosceles Triangle has Two Equal Angles",
"Angles on Equal Arcs are Equal",
"Angles on Equal Arcs are Equal",
"Two Straight Lines make Equal Opposite Angles",
"Definition:Angle",
"Triangle with Two... |
proofwiki-21468 | Broken Chord Theorem | :300px
Let $A$ and $C$ be arbitrary points on a circle in the plane.
Let $M$ be a point on the circle with arc $AM = $ arc $MC$.
Let $B$ lie on the minor arc of $AM$.
Draw chords $AB$ and $BC$.
Find $D$ such that $MD \perp BC$.
Then:
:$AB + BD = DC$ | 300px
Given $MD \perp BC$
Draw $MN \parallel BC$ to meet the circle at $N$.
Draw $NE \parallel MD$.
By Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel:
:$MNED$ is a parallelogram
By Parallelogram with One Right Angle is Rectangle:
:$MNED$ is a rectangle.
{{begin-eqn}}
{{eqn | l = D... | :[[File:Broken Chord.png|300px]]
Let $A$ and $C$ be arbitrary [[Definition:Point|points]] on a [[Definition:Circle|circle]] in the [[Definition:Cartesian Plane|plane]].
Let $M$ be a point on the circle with [[Definition:Arc of Circle|arc]] $AM = $ [[Definition:Arc of Circle|arc]] $MC$.
Let $B$ lie on the [[Definiti... | [[File:BrokenChordTheorem-5.png|300px]]
Given $MD \perp BC$
Draw $MN \parallel BC$ to meet the [[Definition:Circle|circle]] at $N$.
Draw $NE \parallel MD$.
By [[Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel]]:
:$MNED$ is a [[Definition:Parallelogram|parallelogram]]
By [[Para... | Broken Chord Theorem/Proof 5 | https://proofwiki.org/wiki/Broken_Chord_Theorem | https://proofwiki.org/wiki/Broken_Chord_Theorem/Proof_5 | [
"Broken Chord Theorem",
"Circles",
"Named Theorems"
] | [
"File:Broken Chord.png",
"Definition:Point",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Minor Arc",
"Definition:Circle/Chord"
] | [
"File:BrokenChordTheorem-5.png",
"Definition:Circle",
"Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel",
"Definition:Quadrilateral/Parallelogram",
"Parallelogram with One Right Angle is Rectangle",
"Definition:Quadrilateral/Rectangle",
"Parallelism implies Equal Cor... |
proofwiki-21469 | Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center | Let an arbitrary circle $K$ be drawn in the plane.
Let $A'$ and $P'$ be arbitrary points on $K$.
Let $T$ be an inversive transformation such that:
:the inversion center of $T$ is $O$
:the inversion circle $O$ for $T$ is chosen such that $OA'$ is a diameter of $K$
:the radius of $O$ is $r$.
Let $A$ and $P$ be the images... | The diagrams show the two cases:
:$K$ completely inside the circle on $O$
300px
:or with some points outside
300px
The proof is the same for both cases.
{{begin-eqn}}
{{eqn | l = OP \cdot OP'
| r = OA \cdot OA'
| c = definition of $T$
}}
{{eqn | ll= \leadsto
| l = \dfrac {OP} {OA}
| r = \dfrac {... | Let an arbitrary [[Definition:Circle|circle]] $K$ be drawn in the [[Definition:Cartesian Plane|plane]].
Let $A'$ and $P'$ be arbitrary [[Definition:Point|points]] on $K$.
Let $T$ be an [[Definition:Inversive Transformation|inversive transformation]] such that:
:the [[Definition:Inversion Center|inversion center]] of ... | The diagrams show the two cases:
:$K$ completely [[Definition:Interior (Geometry)|inside]] the [[Definition:Circle|circle]] on $O$
[[File:CircleInverse1.png|300px]]
:or with some [[Definition:Point|points]] [[Definition:Exterior (Geometry)|outside]]
[[File:CircleInverse2.png|300px]]
The proof is the same for bo... | Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center | https://proofwiki.org/wiki/Inverse_of_Circle_Through_Inversion_Center_is_Straight_Line_Not_Through_Inversion_Center | https://proofwiki.org/wiki/Inverse_of_Circle_Through_Inversion_Center_is_Straight_Line_Not_Through_Inversion_Center | [
"Inversive Transformations"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Center",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Diameter",
"Definition:Radius",
"Definition:Image (Set Theory)/Map... | [
"Definition:Interior (Geometry)",
"Definition:Circle",
"File:CircleInverse1.png",
"Definition:Point",
"Definition:Exterior (Geometry)",
"File:CircleInverse2.png",
"Triangles with One Equal Angle and Two Sides Proportional are Similar",
"Perpendicular from Point to Straight Line in Plane is Unique",
... |
proofwiki-21470 | Inverse of Straight Line Not Through Inversion Center is Circle Through Inversion Center | Let $C$ be a circle on center $O$ in the plane.
Let $T : X \to Y$ be an inversive transformation with $C$ as the inversion circle.
Then $O$ is the inversion center.
Let $L$ be an arbitrary straight line not containing $O$.
Let $P$ be an arbitrary point on $L$.
Let $P'$ be the image of $P$ under $T$.
Then $P'$ lies on ... | By Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center:
:the image under $T$ of a circle through $C$ is a straight line not through $C$.
By definition of inversive transformation, $T$ is an involution.
Thus, the image under $T$ of a straight line not through $C$ is a circle through ... | Let $C$ be a [[Definition:Circle|circle]] on [[Definition:Center of Circle|center]] $O$ in [[Definition:The Plane|the plane]].
Let $T : X \to Y$ be an [[Definition:Inversive Transformation|inversive transformation]] with $C$ as the [[Definition:Inversion Circle|inversion circle]].
Then $O$ is the [[Definition:Invers... | By [[Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center]]:
:the [[Definition:Image of Element under Mapping|image]] under $T$ of a [[Definition:Circle|circle]] through $C$ is a [[Definition:Straight Line|straight line]] not through $C$.
By definition of [[Definition:Inversive Tran... | Inverse of Straight Line Not Through Inversion Center is Circle Through Inversion Center | https://proofwiki.org/wiki/Inverse_of_Straight_Line_Not_Through_Inversion_Center_is_Circle_Through_Inversion_Center | https://proofwiki.org/wiki/Inverse_of_Straight_Line_Not_Through_Inversion_Center_is_Circle_Through_Inversion_Center | [
"Inversive Transformations"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Plane Surface/The Plane",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Inversive Transformation/Inversion Center",
"Definition:Line/Straight Line",
"Definition:Point",
"Definiti... | [
"Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Circle",
"Definition:Line/Straight Line",
"Definition:Inversive Transformation",
"Definition:Involution",
"Definition:Image (Set Theory)/Mapping/Elemen... |
proofwiki-21471 | Ring of Bounded Continuous Functions is Ring of Continuous Functions for Compact Space | Let $\struct {K, \tau}$ be a compact space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:$\struct {\map {C^*} {S, \R},... | Follows immediately from:
* Compact Space is Pseudocompact Space
* Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space
{{qed}} | Let $\struct {K, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Let $\struc... | Follows immediately from:
* [[Compact Space is Pseudocompact Space]]
* [[Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space]]
{{qed}} | Ring of Bounded Continuous Functions is Ring of Continuous Functions for Compact Space | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Ring_of_Continuous_Functions_for_Compact_Space | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Ring_of_Continuous_Functions_for_Compact_Space | [
"Rings of Continuous Real-Valued Functions",
"Rings of Bounded Continuous Real-Valued Functions",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Bounded Continuous Real-Valued Functions"
] | [
"Compact Space is Pseudocompact",
"Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space"
] |
proofwiki-21472 | Inverse of Circle Not Through Inversion Center | Let $\CC$ be a circle in the plane on center $O$.
Let $T : X \to Y$ be an inversive transformation with $\CC$ as the inversion circle.
Then $O$ is the inversion center.
Let $K$ be an arbitrary circle distinct from $\CC$ and not through $O$.
Then the image under $T$ of all the points on $K$ lie on the same circle, dist... | Let $AB$ be the diameter of $K$, drawn so that $OAB$ are collinear points.
Let $A'$ and $B'$ be the image of $A$ and $B$ under $T$.
Let $K'$ be the circle with diameter $A'B'$
Let $C$ be an otherwise arbitrary point, lying on $K$.
:400px
{{begin-eqn}}
{{eqn | l = OA \cdot OA'
| r = OB \cdot OB' = OC \cdot OC'
... | Let $\CC$ be a [[Definition:Circle|circle]] in the [[Definition:Plane|plane]] on [[Definition:Center of Circle|center]] $O$.
Let $T : X \to Y$ be an [[Definition:Inversive Transformation|inversive transformation]] with $\CC$ as the [[Definition:Inversion Circle|inversion circle]].
Then $O$ is the [[Definition:Invers... | Let $AB$ be the [[Definition:Diameter of Circle|diameter]] of $K$, drawn so that $OAB$ are [[Definition:Collinear Points|collinear points]].
Let $A'$ and $B'$ be the [[Definition:Image of Element under Mapping|image]] of $A$ and $B$ under $T$.
Let $K'$ be the [[Definition:Circle|circle]] with diameter $A'B'$
Let $C$... | Inverse of Circle Not Through Inversion Center/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Circle_Not_Through_Inversion_Center | https://proofwiki.org/wiki/Inverse_of_Circle_Not_Through_Inversion_Center/Proof_1 | [
"Inverse of Circle Not Through Inversion Center",
"Inversive Transformations"
] | [
"Definition:Circle",
"Definition:Plane Surface",
"Definition:Circle/Center",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Inversive Transformation/Inversion Center",
"Definition:Circle",
"Definition:Image (Set Theory)/Mapping/Element",
"D... | [
"Definition:Circle/Diameter",
"Definition:Collinear/Points",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Circle",
"Definition:Point",
"File:Inverse Proof 4a2.png",
"Triangles with One Equal Angle and Two Sides Proportional are Similar",
"Triangles with One Equal Angle and Two Sides Pr... |
proofwiki-21473 | Inverse of Circle Not Through Inversion Center | Let $\CC$ be a circle in the plane on center $O$.
Let $T : X \to Y$ be an inversive transformation with $\CC$ as the inversion circle.
Then $O$ is the inversion center.
Let $K$ be an arbitrary circle distinct from $\CC$ and not through $O$.
Then the image under $T$ of all the points on $K$ lie on the same circle, dist... | :400px
Let the radius of $K$ be $k$.
Draw an arbitrary straight line from $O$ cutting $K$ at $A$ and $B$.
Let $A'$ be the image of $A$ under $T$.
Let $B'$ be the image of $B$ under $T$.
There are two cases:
:$(1): \quad$ The inversion circle $C$ may pass through straight line $OAB$ outside $K$
:$(2): \quad$ $C$ may... | Let $\CC$ be a [[Definition:Circle|circle]] in the [[Definition:Plane|plane]] on [[Definition:Center of Circle|center]] $O$.
Let $T : X \to Y$ be an [[Definition:Inversive Transformation|inversive transformation]] with $\CC$ as the [[Definition:Inversion Circle|inversion circle]].
Then $O$ is the [[Definition:Invers... | :[[File:Inverse Proof 4b.png|400px]]
Let the [[Definition:Radius|radius]] of $K$ be $k$.
Draw an arbitrary [[Definition:Straight Line|straight line]] from $O$ cutting $K$ at $A$ and $B$.
Let $A'$ be the [[Definition:Image of Element under Mapping|image]] of $A$ under $T$.
Let $B'$ be the [[Definition:Image of Ele... | Inverse of Circle Not Through Inversion Center/Proof 2 | https://proofwiki.org/wiki/Inverse_of_Circle_Not_Through_Inversion_Center | https://proofwiki.org/wiki/Inverse_of_Circle_Not_Through_Inversion_Center/Proof_2 | [
"Inverse of Circle Not Through Inversion Center",
"Inversive Transformations"
] | [
"Definition:Circle",
"Definition:Plane Surface",
"Definition:Circle/Center",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Inversive Transformation/Inversion Center",
"Definition:Circle",
"Definition:Image (Set Theory)/Mapping/Element",
"D... | [
"File:Inverse Proof 4b.png",
"Definition:Radius",
"Definition:Line/Straight Line",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Line/Straight Line",
"Definition:Inversive Transforma... |
proofwiki-21474 | Continuous Real-Valued Function on Compact Space is Bounded | Let $\struct {K, \tau}$ be a compact space.
Let $\R$ denote the real number line.
Let $f: S \to \R$ be a continuous real-valued function.
Then:
:$f$ is bounded. | From Compact Space is Pseudocompact Space:
:$\struct {K, \tau}$ is pseudocompact
By definition of pseudocompact:
:$f$ is bounded.
{{qed}} | Let $\struct {K, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $f: S \to \R$ be a [[Definition:Continuous Real-Valued Function|continuous real-valued function]].
Then:
:$f$ is [[Definition:Bounded Real-Valued Function|bo... | From [[Compact Space is Pseudocompact Space]]:
:$\struct {K, \tau}$ is [[Definition:Pseudocompact Space|pseudocompact]]
By definition of [[Definition:Pseudocompact Space|pseudocompact]]:
:$f$ is [[Definition:Bounded Real-Valued Function|bounded]].
{{qed}} | Continuous Real-Valued Function on Compact Space is Bounded | https://proofwiki.org/wiki/Continuous_Real-Valued_Function_on_Compact_Space_is_Bounded | https://proofwiki.org/wiki/Continuous_Real-Valued_Function_on_Compact_Space_is_Bounded | [
"Bounded Real-Valued Functions",
"Continuous Real-Valued Functions",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Continuous Real-Valued Vector Function",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Compact Space is Pseudocompact",
"Definition:Pseudocompact Space",
"Definition:Pseudocompact Space",
"Definition:Bounded Mapping/Real-Valued"
] |
proofwiki-21475 | Compact Space is Pseudocompact | Let $\struct {K, \tau}$ be a compact space.
Then $\struct {K, \tau}$ is a pseudocompact space | Follows immediately from:
* Compact Space is Countably Compact
* Countably Compact Space is Pseudocompact
{{qed}} | Let $\struct {K, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $\struct {K, \tau}$ is a [[Definition:Pseudocompact Space|pseudocompact space]] | Follows immediately from:
* [[Compact Space is Countably Compact]]
* [[Countably Compact Space is Pseudocompact]]
{{qed}} | Compact Space is Pseudocompact | https://proofwiki.org/wiki/Compact_Space_is_Pseudocompact | https://proofwiki.org/wiki/Compact_Space_is_Pseudocompact | [
"Pseudocompact Spaces",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Pseudocompact Space"
] | [
"Compact Space is Countably Compact",
"Countably Compact Space is Pseudocompact"
] |
proofwiki-21476 | Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space | Let $\struct {K, \tau}$ be a pseudocompact space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.
Then:
:$\struct {\map {C^*} {S... | Follows immediately from the definitions of:
* Definition:Pseudocompact Space
* Definition:Ring of Continuous Real-Valued Functions
* Definition:Ring of Bounded Continuous Real-Valued Functions
{{qed}} | Let $\struct {K, \tau}$ be a [[Definition:Pseudocompact Space|pseudocompact space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\map C {S, \R}, +, *}$ be the [[Definition:Ring of Continuous Real-Valued Functions|ring of continuous real-valued functions]] from $S$.
Let $\struc... | Follows immediately from the definitions of:
* [[Definition:Pseudocompact Space]]
* [[Definition:Ring of Continuous Real-Valued Functions]]
* [[Definition:Ring of Bounded Continuous Real-Valued Functions]]
{{qed}} | Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Ring_of_Continuous_Functions_for_Pseudocompact_Space | https://proofwiki.org/wiki/Ring_of_Bounded_Continuous_Functions_is_Ring_of_Continuous_Functions_for_Pseudocompact_Space | [
"Rings of Continuous Real-Valued Functions",
"Rings of Bounded Continuous Real-Valued Functions",
"Pseudocompact Spaces"
] | [
"Definition:Pseudocompact Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Pseudocompact Space",
"Definition:Ring of Continuous Real-Valued Functions",
"Definition:Ring of Bounded Continuous Real-Valued Functions"
] |
proofwiki-21477 | Lattice of Real-Valued Functions forms Distributive Lattice | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\R^S, \vee, \wedge, \le}$ be the lattice of real-valued functions from $S$ where:
:$\forall f, g \in \R^S : f \vee g : S \to \R$ is defined by:
::$\forall s \in S : \map {\paren{f \vee g}} s = \max \set{\map f s, \... | From Usual Ordering on Real Numbers is Total Ordering:
:$\struct {\R, \le}$ is totally ordered
where $\le$ denotes the usual ordering on $\R$.
From Totally Ordered Set is Lattice
:$\struct{\R, \le}$ is a lattice.
By definition of join:
:$\forall x, y \in \R : x \vee y = \sup \set {x, y}$
where $x \vee y$ denotes the jo... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\R^S, \vee, \wedge, \le}$ be the [[Definition:Lattice of Real-Valued Functions|lattice of real-valued functions]] from $S$ where:
:$\forall f, g ... | From [[Usual Ordering on Real Numbers is Total Ordering]]:
:$\struct {\R, \le}$ is [[Definition:Total Ordering|totally ordered]]
where $\le$ denotes the [[Definition:Usual Ordering|usual ordering]] on $\R$.
From [[Totally Ordered Set is Lattice]]
:$\struct{\R, \le}$ is a [[Definition:Lattice (Order Theory)|lattice]].... | Lattice of Real-Valued Functions forms Distributive Lattice | https://proofwiki.org/wiki/Lattice_of_Real-Valued_Functions_forms_Distributive_Lattice | https://proofwiki.org/wiki/Lattice_of_Real-Valued_Functions_forms_Distributive_Lattice | [
"Distributive Lattices",
"Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Lattice of Real-Valued Functions",
"Definition:Distributive Lattice"
] | [
"Total Ordering/Examples/Usual Ordering on Real Numbers",
"Definition:Total Ordering",
"Definition:Usual Ordering",
"Totally Ordered Set is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Join (Order Theory)",
"Definition:Join (Order Theory)",
"Max Operation Yields Supremum of Parameters",
... |
proofwiki-21478 | Structure Induced by Idempotent Operation is Idempotent | Let $\struct {T, \circ}$ be an algebraic structure, and let $S$ be a set.
Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.
Let $\circ$ be an idempotent operation.
Then the pointwise operation $\oplus$ induced on $T^S$ by $\circ$ is also idempotent. | Let $f \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {f \oplus f} } x
| r = \map f x \circ \map f x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x
| c = $\circ$ is idempotent operation
}}
{{end-eqn}}
From Equality of Mappings:
:$f \oplus f = f$
Since ... | Let $\struct {T, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]], and let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $\circ$.
Let $\circ$ be an [[Definition:Idempotent Operation|idempotent o... | Let $f \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {f \oplus f} } x
| r = \map f x \circ \map f x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x
| c = $\circ$ is [[Definition:Idempotent Operation|idempotent operation]]
}}
{{end-eqn}}
From [[Equ... | Structure Induced by Idempotent Operation is Idempotent | https://proofwiki.org/wiki/Structure_Induced_by_Idempotent_Operation_is_Idempotent | https://proofwiki.org/wiki/Structure_Induced_by_Idempotent_Operation_is_Idempotent | [
"Pointwise Operations",
"Idempotence"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Idempotence/Operation",
"Definition:Pointwise Operation",
"Definition:Idempotence/Operation"
] | [
"Definition:Idempotence/Operation",
"Equality of Mappings",
"Definition:Idempotence/Operation",
"Category:Pointwise Operations",
"Category:Idempotence"
] |
proofwiki-21479 | Structure Induced by Lattice Operations is Lattice | Let $\struct {L, \vee, \wedge, \preceq}$ be a lattice.
Let $S$ be a set.
Let $\struct {L^S, \veebar, \barwedge}$ be the structure on $T^S$ induced by $\vee$ and $\wedge$.
Let $\precsim$ be the ordering on $S$ defined by:
:$\forall f, g \in L^S: f \precsim g$ {{iff}} $f \veebar g = g$
as on Semilattice Induces Ordering.... | === $\struct {L^S, \veebar}$ and $\struct {L^S, \barwedge}$ are Semilattices ===
By definition of lattice:
:$\struct {L, \vee}$ and $\struct {L, \wedge}$ are semilattices.
From Structure Induced by Semilattice Operation is Semilattice:
:$\struct {L^S, \veebar}$ and $\struct {L^S, \veebar}$ are semilattices.
{{qed|lemma... | Let $\struct {L, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $S$ be a [[Definition:Set|set]].
Let $\struct {L^S, \veebar, \barwedge}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $\vee$ and $\wedge$.
Let $\precsim$ be the [[Definition:Ordering|ordering]]... | === $\struct {L^S, \veebar}$ and $\struct {L^S, \barwedge}$ are Semilattices ===
By definition of [[Definition:Lattice (Order Theory)|lattice]]:
:$\struct {L, \vee}$ and $\struct {L, \wedge}$ are [[Definition:Semilattice|semilattices]].
From [[Structure Induced by Semilattice Operation is Semilattice]]:
:$\struct {L... | Structure Induced by Lattice Operations is Lattice | https://proofwiki.org/wiki/Structure_Induced_by_Lattice_Operations_is_Lattice | https://proofwiki.org/wiki/Structure_Induced_by_Lattice_Operations_is_Lattice | [
"Lattices (Order Theory)",
"Pointwise Operations"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Ordering",
"Semilattice Induces Ordering",
"Definition:Lattice (Order Theory)"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Semilattice",
"Structure Induced by Semilattice Operation is Semilattice",
"Definition:Semilattice",
"Definition:Lattice (Order Theory)",
"Definition:Lattice (Order Theory)"
] |
proofwiki-21480 | Structure Induced by Absorbing Operations is Absorbing | Let $\struct {T, \circ, *}$ be an algebraic structure, and let $S$ be a set.
Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $\circ$ and $*$.
Let $\circ$ and $*$ satisfy the absorption law:
:$\forall a, b \in S: a \circ \paren {a * b} = a$
Then the pointwise operations $\oplus$ and $\otimes$ o... | Let $f, g \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {f \oplus \paren {f \otimes g} } } x
| r = \map f x \circ \map {\paren {f \otimes g} } x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x \circ \paren {\map f x * \map g x}
| c = {{Defof|Pointwise O... | Let $\struct {T, \circ, *}$ be an [[Definition:Algebraic Structure with Two Operations|algebraic structure]], and let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus, \otimes}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $\circ$ and $*$.
Let $\circ$ and $*$ satisfy the [[Definitio... | Let $f, g \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {f \oplus \paren {f \otimes g} } } x
| r = \map f x \circ \map {\paren {f \otimes g} } x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x \circ \paren {\map f x * \map g x}
| c = {{Defof|Pointwis... | Structure Induced by Absorbing Operations is Absorbing | https://proofwiki.org/wiki/Structure_Induced_by_Absorbing_Operations_is_Absorbing | https://proofwiki.org/wiki/Structure_Induced_by_Absorbing_Operations_is_Absorbing | [
"Absorption Laws",
"Pointwise Operations"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Absorption Law",
"Definition:Pointwise Operation",
"Definition:Absorption Law"
] | [
"Definition:Absorption Law",
"Equality of Mappings",
"Definition:Absorption Law",
"Category:Absorption Laws",
"Category:Pointwise Operations"
] |
proofwiki-21481 | Structure Induced by Semilattice Operation is Semilattice | Let $\struct {T, \circ}$ be a semilattice.
Let $S$ be a set.
Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.
Then $\struct {T^S, \oplus}$ is a semilattice. | Taking the semilattice axioms in turn: | Let $\struct {T, \circ}$ be a [[Definition:Semilattice|semilattice]].
Let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $\circ$.
Then $\struct {T^S, \oplus}$ is a [[Definition:Semilattice|semilattice]]. | Taking the [[Axiom:Semilattice Axioms|semilattice axioms]] in turn: | Structure Induced by Semilattice Operation is Semilattice | https://proofwiki.org/wiki/Structure_Induced_by_Semilattice_Operation_is_Semilattice | https://proofwiki.org/wiki/Structure_Induced_by_Semilattice_Operation_is_Semilattice | [
"Semilattices",
"Pointwise Operations"
] | [
"Definition:Semilattice",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Semilattice"
] | [
"Axiom:Semilattice Axioms",
"Axiom:Semilattice Axioms"
] |
proofwiki-21482 | Perpendicular from Point to Straight Line in Plane is Unique | Let $BC$ be a straight line in the plane.
Let $AM$ be perpendicular to $BC$ at $M$.
Let $P$ be an arbitrary point on the same side of $BC$ as $A$ is, but not on $AM$.
Then $PM$ cannot be perpendicular to $BC$. | :250px
{{WLOG}} let $M$ bisect $BC$.
{{AimForCont}} $PM \perp BC$.
By definition $PM$ is a perpendicular bisector of $BC$ at $M$.
Then by definition of perpendicular bisector:
:$PC = PB$
Find $P'$ on $AM$ such that $P'C = PC$.
Since $AM$ is also a perpendicular bisector of $BC$:
:$P'B = P'C$
:$\leadsto PB = PC = P'C = ... | Let $BC$ be a [[Definition:Straight Line|straight line]] in the plane.
Let $AM$ be [[Definition:Perpendicular|perpendicular]] to $BC$ at $M$.
Let $P$ be an arbitrary [[Definition:Point|point]] on the same side of $BC$ as $A$ is, but not on $AM$.
Then $PM$ cannot be [[Definition:Perpendicular|perpendicular]] to $BC$. | :[[File:Perp bisector.png|250px]]
{{WLOG}} let $M$ [[Definition:Bisection|bisect]] $BC$.
{{AimForCont}} $PM \perp BC$.
By definition $PM$ is a [[Definition:Perpendicular Bisector|perpendicular bisector]] of $BC$ at $M$.
Then by definition of [[Definition:Perpendicular Bisector|perpendicular bisector]]:
:$PC = PB$
... | Perpendicular from Point to Straight Line in Plane is Unique | https://proofwiki.org/wiki/Perpendicular_from_Point_to_Straight_Line_in_Plane_is_Unique | https://proofwiki.org/wiki/Perpendicular_from_Point_to_Straight_Line_in_Plane_is_Unique | [
"Perpendicular from Point to Straight Line in Plane is Unique",
"Perpendiculars",
"Planes"
] | [
"Definition:Line/Straight Line",
"Definition:Right Angle/Perpendicular",
"Definition:Point",
"Definition:Right Angle/Perpendicular"
] | [
"File:Perp bisector.png",
"Definition:Bisection",
"Definition:Perpendicular Bisector",
"Definition:Perpendicular Bisector",
"Definition:Perpendicular Bisector",
"Definition:Contradiction",
"Definition:Point",
"Definition:Equidistance"
] |
proofwiki-21483 | Products of Homeomorphic Spaces are Homeomorphic | Let:
:$\sequence {T_i}_{i \mathop \in I}$
:$\sequence {T'_i}_{i \mathop \in I}$
be indexed families of topological spaces, with indexing set $I$.
Let:
:$\sequence {\phi_i}_{i \mathop \in I}$
be an indexed family of homeomorphisms $\phi_i$ from $T_i$ to $T'_i$.
Define:
:$\ds T = \prod_{i \mathop \in I} T_i$
:$\ds T' = \... | We have that:
:$\map {\phi^{-1}} x = \sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I}$
For:
{{begin-eqn}}
{{eqn | l = \map \phi {\map {\phi^{-1} } x}
| r = \sequence {\map {\phi_i \circ \pr_i} {\sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I} } }_{i \mathop \in I}
}}
{{eqn | r = \seque... | Let:
:$\sequence {T_i}_{i \mathop \in I}$
:$\sequence {T'_i}_{i \mathop \in I}$
be [[Definition:Indexed Family|indexed families]] of [[Definition:Topological Space|topological spaces]], with [[Definition:Indexing Set|indexing set]] $I$.
Let:
:$\sequence {\phi_i}_{i \mathop \in I}$
be an [[Definition:Indexed Family|ind... | We have that:
:$\map {\phi^{-1}} x = \sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I}$
For:
{{begin-eqn}}
{{eqn | l = \map \phi {\map {\phi^{-1} } x}
| r = \sequence {\map {\phi_i \circ \pr_i} {\sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I} } }_{i \mathop \in I}
}}
{{eqn | r = \sequ... | Products of Homeomorphic Spaces are Homeomorphic | https://proofwiki.org/wiki/Products_of_Homeomorphic_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Products_of_Homeomorphic_Spaces_are_Homeomorphic | [
"Homeomorphisms (Topological Spaces)",
"Product Topology"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Product Space (Topology)",
"Definition:Mapping",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Bijection",
"Projection from Product Topology is Continuous/General Result",
"Composite of Continuous Mappings is Continuous",
"Continuous Mapping to Product Space/General Result",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Homeomorphism/Topological Spaces",
"Category... |
proofwiki-21484 | Rotation in Euclidean Space is Conformal | Let $f$ be an rotation in Euclidean space.
Then $f$ is a conformal transformation. | {{ProofWanted}}
Category:Geometric Rotations
Category:Conformal Transformations
s2rtr95rl73qb00l4o4qczb1lf2yn99 | Let $f$ be an [[Definition:Plane Rotation|rotation]] in [[Definition:Euclidean Space|Euclidean space]].
Then $f$ is a [[Definition:Conformal Transformation|conformal transformation]]. | {{ProofWanted}}
[[Category:Geometric Rotations]]
[[Category:Conformal Transformations]]
s2rtr95rl73qb00l4o4qczb1lf2yn99 | Rotation in Euclidean Space is Conformal | https://proofwiki.org/wiki/Rotation_in_Euclidean_Space_is_Conformal | https://proofwiki.org/wiki/Rotation_in_Euclidean_Space_is_Conformal | [
"Geometric Rotations",
"Conformal Transformations"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Euclidean Space",
"Definition:Conformal Transformation"
] | [
"Category:Geometric Rotations",
"Category:Conformal Transformations"
] |
proofwiki-21485 | Lattice and Ring of Real-Valued Functions forms Ordered Ring | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\R^S, +, \times}$ be the ring of real-valued functions from $S$ to $\R$.
Let $\struct {\R^S, \vee, \wedge, \le}$ be the lattice of real-valued functions from $S$ to $\R$.
Then:
:$\struct {\R^S, +, \times, \le}$ is ... | From Structure Induced by Group Operation is Group:
:the zero of $\struct {\R^S, +, \times}$ is the constant mapping $0_{R^S} : S \to R$ defined by:
::$\forall s \in S : \map {0_{R^S}} s = 0_R$
It needs to be shown that the order $\le$ on the lattice of real-valued functions satisies the ring compatible ordering axioms... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\R^S, +, \times}$ be the [[Definition:Ring of Mappings|ring of real-valued functions]] from $S$ to $\R$.
Let $\struct {\R^S, \vee, \wedge, \le}$ ... | From [[Structure Induced by Group Operation is Group]]:
:the [[Definition:Ring Zero|zero]] of $\struct {\R^S, +, \times}$ is the [[Definition:Constant Mapping|constant mapping]] $0_{R^S} : S \to R$ defined by:
::$\forall s \in S : \map {0_{R^S}} s = 0_R$
It needs to be shown that the [[Definition:Ordering|order]] $\l... | Lattice and Ring of Real-Valued Functions forms Ordered Ring | https://proofwiki.org/wiki/Lattice_and_Ring_of_Real-Valued_Functions_forms_Ordered_Ring | https://proofwiki.org/wiki/Lattice_and_Ring_of_Real-Valued_Functions_forms_Ordered_Ring | [
"Real-Valued Functions",
"Ordered Rings"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Ring of Mappings",
"Definition:Lattice of Real-Valued Functions",
"Definition:Ordered Ring"
] | [
"Structure Induced by Group Operation is Group",
"Definition:Ring Zero",
"Definition:Constant Mapping",
"Definition:Ordering",
"Definition:Lattice of Real-Valued Functions",
"Axiom:Ring Compatible Ordering Axioms",
"Definition:Relation Compatible with Operation",
"Definition:Ring (Abstract Algebra)/Pr... |
proofwiki-21486 | Eccentricity of Parabola equals 1 | The parabola has eccentricity equal to $1$. | Let $K$ be a '''conic section'''.
From the focus-directrix property of a conic section, $K$ is the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition:
:$(1): \quad q = \epsilon p$
where $\epsilon$ denotes the eccentricity.
Now let $K$ be a pa... | The [[Definition:Parabola|parabola]] has [[Definition:Eccentricity of Conic Section|eccentricity]] equal to $1$. | Let $K$ be a '''[[Definition:Conic Section|conic section]]'''.
From the [[Definition:Focus-Directrix Property of Conic Section|focus-directrix property of a conic section]], $K$ is the [[Definition:Locus|locus]] of [[Definition:Point|points]] $b$ such that the [[Definition:Length (Linear Measure)|distance]] $p$ from $... | Eccentricity of Parabola equals 1 | https://proofwiki.org/wiki/Eccentricity_of_Parabola_equals_1 | https://proofwiki.org/wiki/Eccentricity_of_Parabola_equals_1 | [
"Parabolas",
"Eccentricity of Conic Section"
] | [
"Definition:Parabola",
"Definition:Conic Section/Eccentricity"
] | [
"Definition:Conic Section",
"Definition:Conic Section/Focus-Directrix Property",
"Definition:Locus",
"Definition:Point",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Propositional Function",
"Definition:Conic Section/Eccentricity",
"Definition:Parabola",
"Def... |
proofwiki-21487 | Value of Discriminant of Conic Section | Let $K$ be a conic section embedded in a Cartesian plane with the general equation:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
where $a, b, c, f, g, h \in \R$.
The value of the discriminant of $K$ is:
:$\Delta = a b c + 2 f g h - a f^2 - b g^2 - c h^2$ | By definition, the discriminant of $K$ is:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}
| c = {{Defof|Discriminant of Conic Section}}
}}
{{eqn | r = a \paren {b c - f^2} - h \paren {h c - g f} + g \paren {h f - g b}
| c = Laplace Expansion ... | Let $K$ be a [[Definition:Conic Section|conic section]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] with the general [[Definition:Equation|equation]]:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
where $a, b, c, f, g, h \in \R$.
The value of the [[Definition:Discriminant of Conic Section|discri... | By definition, the [[Definition:Discriminant of Conic Section|discriminant]] of $K$ is:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}
| c = {{Defof|Discriminant of Conic Section}}
}}
{{eqn | r = a \paren {b c - f^2} - h \paren {h c - g f} + g \pa... | Value of Discriminant of Conic Section | https://proofwiki.org/wiki/Value_of_Discriminant_of_Conic_Section | https://proofwiki.org/wiki/Value_of_Discriminant_of_Conic_Section | [
"Discriminants of Conic Sections"
] | [
"Definition:Conic Section",
"Definition:Cartesian Plane",
"Definition:Equation",
"Definition:Discriminant of Conic Section"
] | [
"Definition:Discriminant of Conic Section",
"Laplace Expansion Theorem for Determinants",
"Distributive Laws/Arithmetic"
] |
proofwiki-21488 | Lattice of Continuous Functions is Sublattice of All Real-Valued Functions | Let $\struct {S, \tau }$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\R^S, \vee, \wedge, \le}$ be the lattice of real-valued functions from $S$.
Let $\struct{\map C {S, R}, \vee, \wedge, \le}$ be the lattice of continuous real-valued functions from $S$.
Then:
:$\struct{\map C {S, R}, \ve... | To show that $\struct{\map C {S, R}, \vee, \wedge, \le}$ is a sublattice of $\struct {\R^S, \vee, \wedge, \le}$ it is sufficient to show that $\map C {S, R}$ is closed under $\vee$ and $\wedge$.
From Maximum Rule for Continuous Real-Valued Functions:
:$\map C {S, R}$ is closed under $\vee$
From Minimum Rule for Continu... | Let $\struct {S, \tau }$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct {\R^S, \vee, \wedge, \le}$ be the [[Definition:Lattice of Real-Valued Functions|lattice of real-valued functions]] from $S$.
Let $\struct{\map C {S, R},... | To show that $\struct{\map C {S, R}, \vee, \wedge, \le}$ is a [[Definition:Sublattice|sublattice]] of $\struct {\R^S, \vee, \wedge, \le}$ it is sufficient to show that $\map C {S, R}$ is [[Definition:Closed Algebraic Structure|closed]] under $\vee$ and $\wedge$.
From [[Maximum Rule for Continuous Real-Valued Function... | Lattice of Continuous Functions is Sublattice of All Real-Valued Functions | https://proofwiki.org/wiki/Lattice_of_Continuous_Functions_is_Sublattice_of_All_Real-Valued_Functions | https://proofwiki.org/wiki/Lattice_of_Continuous_Functions_is_Sublattice_of_All_Real-Valued_Functions | [
"Lattices of Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Lattice of Real-Valued Functions",
"Definition:Lattice of Continuous Real-Valued Functions",
"Definition:Sublattice"
] | [
"Definition:Sublattice",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Combination Theorem for Continuous Real-Valued Functions/Maximum Rule",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Combination Theorem for Continuous Real-Valued Functions/Minimum Rule",
"Definition:C... |
proofwiki-21489 | Classification of Conic Sections by Coefficients of General Equation | Let $K$ be a conic section embedded in a Cartesian plane with the general equation:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
where $a, b, c, f, g, h \in \R$.
Let $\Delta$ denote the discriminant of $K$:
:$\Delta = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$
If $\Delta \ne 0$, then $K$... | {{ProofWanted|a satisfying job for a rainy day}} | Let $K$ be a [[Definition:Conic Section|conic section]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] with the general [[Definition:Equation|equation]]:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
where $a, b, c, f, g, h \in \R$.
Let $\Delta$ denote the [[Definition:Discriminant of Conic Section... | {{ProofWanted|a satisfying job for a rainy day}} | Classification of Conic Sections by Coefficients of General Equation | https://proofwiki.org/wiki/Classification_of_Conic_Sections_by_Coefficients_of_General_Equation | https://proofwiki.org/wiki/Classification_of_Conic_Sections_by_Coefficients_of_General_Equation | [
"Conic Sections"
] | [
"Definition:Conic Section",
"Definition:Cartesian Plane",
"Definition:Equation",
"Definition:Discriminant of Conic Section",
"Definition:Ellipse",
"Definition:Parabola",
"Definition:Hyperbola",
"Definition:Ellipse",
"Definition:Parabola",
"Definition:Hyperbola",
"Definition:Degenerate Conic",
... | [] |
proofwiki-21490 | Lattice of Bounded Continuous Functions is Sublattice of Continuous Real-Valued Functions | Let $\struct {S, \tau }$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct{\map C {S, R}, \vee, \wedge, \le}$ be the lattice of continuous real-valued functions from $S$.
Let $\struct{\map {C^*} {S, R}, \vee, \wedge, \le}$ be the lattice of bounded continuous real-valued functions from $S$.
The... | To show that $\struct{\map {C^*} {S, R}, \vee, \wedge, \le}$ is a sublattice of $\struct {\map C {S, R}, \vee, \wedge, \le}$ it is sufficient to show that $\map {C^*} {S, R}$ is closed under $\vee$ and $\wedge$.
From Maximum Rule for Bounded Continuous Real-Valued Functions:
:$\map {C^*} {S, R}$ is closed under $\vee$
... | Let $\struct {S, \tau }$ be a [[Definition:Topological Space|topological space]].
Let $\R$ denote the [[Definition:Real Number Line|real number line]].
Let $\struct{\map C {S, R}, \vee, \wedge, \le}$ be the [[Definition:Lattice of Continuous Real-Valued Functions|lattice of continuous real-valued functions]] from $S$... | To show that $\struct{\map {C^*} {S, R}, \vee, \wedge, \le}$ is a [[Definition:Sublattice|sublattice]] of $\struct {\map C {S, R}, \vee, \wedge, \le}$ it is sufficient to show that $\map {C^*} {S, R}$ is [[Definition:Closed Algebraic Structure|closed]] under $\vee$ and $\wedge$.
From [[Maximum Rule for Bounded Contin... | Lattice of Bounded Continuous Functions is Sublattice of Continuous Real-Valued Functions | https://proofwiki.org/wiki/Lattice_of_Bounded_Continuous_Functions_is_Sublattice_of_Continuous_Real-Valued_Functions | https://proofwiki.org/wiki/Lattice_of_Bounded_Continuous_Functions_is_Sublattice_of_Continuous_Real-Valued_Functions | [
"Lattices of Bounded Continuous Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real Number/Real Number Line",
"Definition:Lattice of Continuous Real-Valued Functions",
"Definition:Lattice of Bounded Continuous Real-Valued Functions",
"Definition:Sublattice"
] | [
"Definition:Sublattice",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Combination Theorem for Bounded Continuous Real-Valued Functions/Maximum Rule",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Combination Theorem for Bounded Continuous Real-Valued Functions/Minimum Rule",... |
proofwiki-21491 | Projection of Conic Section is Conic Section | Let $K$ be a conic section.
Let $K'$ be a projection of $K$.
Then $K'$ is also a conic section. | {{ProofWanted|needs background work completed}} | Let $K$ be a [[Definition:Conic Section|conic section]].
Let $K'$ be a [[Definition:Geometric Projection|projection]] of $K$.
Then $K'$ is also a [[Definition:Conic Section|conic section]]. | {{ProofWanted|needs background work completed}} | Projection of Conic Section is Conic Section | https://proofwiki.org/wiki/Projection_of_Conic_Section_is_Conic_Section | https://proofwiki.org/wiki/Projection_of_Conic_Section_is_Conic_Section | [
"Conic Sections",
"Projective Geometry"
] | [
"Definition:Conic Section",
"Definition:Projection (Geometry)",
"Definition:Conic Section"
] | [] |
proofwiki-21492 | Power of a Point Theorem | Let $C$ be a circle in the Euclidean plane whose center is $O$ and whose radius is $r$.
Let $P$ be an arbitrary point in the plane.
Let $p$ be the power of point $P$ {{WRT}} $C$.
Let a directed line segment from $P$ be drawn either:
:intersecting $C$ at two points $A$ and $A'$
or:
:tangent to $C$ at $A = A'$.
Then:
:$P... | Let $d$ be the distance from $P$ to $O$.
Let $t$ be the length of the tangent from $P$ to $C$.
We use the following several times.
{{begin-eqn}}
{{eqn | l = PA
| r = -AP
| c = {{Defof|Directed Line Segment}}
}}
{{eqn | n = 1
| l = PA \cdot PA'
| r = -AP \cdot PA'
| c = substitution
}}
{{... | Let $C$ be a [[Definition:Circle|circle]] in the [[Definition:Euclidean Plane|Euclidean plane]] whose [[Definition:Center of Circle|center]] is $O$ and whose [[Definition:Radius of Circle|radius]] is $r$.
Let $P$ be an arbitrary [[Definition:Point|point]] in the [[Definition:Euclidean Plane|plane]].
Let $p$ be the [[... | Let $d$ be the [[Definition:Distance between Points|distance]] from $P$ to $O$.
Let $t$ be the [[Definition:Length (Linear Measure)|length]] of the [[Definition:Tangent|tangent]] from $P$ to $C$.
We use the following several times.
{{begin-eqn}}
{{eqn | l = PA
| r = -AP
| c = {{Defof|Directed Line Segme... | Power of a Point Theorem | https://proofwiki.org/wiki/Power_of_a_Point_Theorem | https://proofwiki.org/wiki/Power_of_a_Point_Theorem | [
"Power of a Point Theorem",
"Power of Point",
"Circles",
"Named Theorems"
] | [
"Definition:Circle",
"Definition:Euclidean Plane",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Euclidean Plane",
"Definition:Power of Point",
"Definition:Directed Line Segment",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Tang... | [
"Definition:Distance between Points",
"Definition:Linear Measure/Length",
"Definition:Tangent",
"Pythagoras's Theorem",
"Definition:Circle/Center",
"Definition:Line/Straight Line",
"Definition:Circle/Diameter",
"File:Power of Point Interior.png",
"Definition:Interior",
"Definition:Circle/Diameter"... |
proofwiki-21493 | Real Number Line less Zero is Disconnected Space | Let $S := \R \setminus \set 0$ be the real number line with $0$ excluded.
Let $\struct {S, \tau_d}$ be $S$ with the usual (Euclidean) topology.
Then $\struct {S, \tau_d}$ is disconnected. | We note that:
:$S = \openint \gets 0 \cup \openint 0 \to$
:$\openint \gets 0 \cap \openint 0 \to = \O$
Hence we have partitioned $S$ into $2$ disjoint open sets whose union is $S$.
Hence by definition $S$ is not connected.
Hence the result by definition of disconnected space.
{{qed}} | Let $S := \R \setminus \set 0$ be the [[Definition:Real Number Line|real number line]] with $0$ excluded.
Let $\struct {S, \tau_d}$ be $S$ with the [[Definition:Real Number Line with Euclidean Topology|usual (Euclidean) topology]].
Then $\struct {S, \tau_d}$ is [[Definition:Disconnected Space|disconnected]]. | We note that:
:$S = \openint \gets 0 \cup \openint 0 \to$
:$\openint \gets 0 \cap \openint 0 \to = \O$
Hence we have [[Definition:Set Partition|partitioned]] $S$ into $2$ [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] whose [[Definition:Set Union|union]] is $S$.
Hence by definitio... | Real Number Line less Zero is Disconnected Space | https://proofwiki.org/wiki/Real_Number_Line_less_Zero_is_Disconnected_Space | https://proofwiki.org/wiki/Real_Number_Line_less_Zero_is_Disconnected_Space | [
"Disconnected Spaces",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Disconnected (Topology)/Topological Space"
] | [
"Definition:Set Partition",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Connected Topological Space",
"Definition:Disconnected (Topology)/Topological Space"
] |
proofwiki-21494 | Positive Real Number has Simple Continued Fraction Expansion | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Then $x$ can be expressed as a simple continued fraction. | We have that $x$ is either rational or irrational.
;$x$ rational
Let $x$ be rational.
Then from Rational Number can be Expressed as Simple Finite Continued Fraction, $x$ has a simple continued fraction expansion.
;$x$ irrational
Let $x$ be irrational.
The result follows from Correspondence between Irrational Numbers an... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then $x$ can be expressed as a [[Definition:Simple Continued Fraction|simple continued fraction]]. | We have that $x$ is either [[Definition:Rational Number|rational]] or [[Definition:Irrational Number|irrational]].
;$x$ [[Definition:Rational Number|rational]]
Let $x$ be [[Definition:Rational Number|rational]].
Then from [[Rational Number can be Expressed as Simple Finite Continued Fraction]], $x$ has a [[Definiti... | Positive Real Number has Simple Continued Fraction Expansion | https://proofwiki.org/wiki/Positive_Real_Number_has_Simple_Continued_Fraction_Expansion | https://proofwiki.org/wiki/Positive_Real_Number_has_Simple_Continued_Fraction_Expansion | [
"Simple Continued Fractions"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Simple Continued Fraction"
] | [
"Definition:Rational Number",
"Definition:Irrational Number",
"Definition:Rational Number",
"Definition:Rational Number",
"Rational Number can be Expressed as Simple Finite Continued Fraction",
"Definition:Simple Continued Fraction",
"Definition:Irrational Number",
"Definition:Irrational Number",
"C... |
proofwiki-21495 | Parallelogram with One Right Angle is Rectangle | Let $\Box ABCD$ be a parallelogram.
Let one angle of $\Box ABCD$ be a right angle.
Then $\Box ABCD$ is a rectangle. | {{WLOG}}, let $\angle ABC$ be a right angle.
Let $\angle CDA$ be opposite $\angle ABC$.
By Opposite Sides and Angles of Parallelogram are Equal:
:$\angle CDA = \angle ABC$
By Sum of Internal Angles of Polygon:
: the sum of the angles of $\Box ABCD$ is four right angles.
Hence:
$\angle DAB + \angle BCD$ together equal t... | Let $\Box ABCD$ be a [[Definition:Parallelogram|parallelogram]].
Let one [[Definition:Angle|angle]] of $\Box ABCD$ be a [[Definition:Right Angle|right angle]].
Then $\Box ABCD$ is a [[Definition:Rectangle|rectangle]]. | {{WLOG}}, let $\angle ABC$ be a [[Definition:Right Angle|right angle]].
Let $\angle CDA$ be [[Definition:Opposite (in Polygon)|opposite]] $\angle ABC$.
By [[Opposite Sides and Angles of Parallelogram are Equal]]:
:$\angle CDA = \angle ABC$
By [[Sum of Internal Angles of Polygon]]:
: the sum of the [[Definition:Angle... | Parallelogram with One Right Angle is Rectangle | https://proofwiki.org/wiki/Parallelogram_with_One_Right_Angle_is_Rectangle | https://proofwiki.org/wiki/Parallelogram_with_One_Right_Angle_is_Rectangle | [
"Parallelograms",
"Rectangles"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Angle",
"Definition:Right Angle",
"Definition:Quadrilateral/Rectangle"
] | [
"Definition:Right Angle",
"Definition:Polygon/Opposite",
"Opposite Sides and Angles of Parallelogram are Equal",
"Sum of Internal Angles of Polygon",
"Definition:Angle",
"Definition:Right Angle",
"Definition:Right Angle",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Angle",
... |
proofwiki-21496 | Structure Induced by Left Distributive Operation is Left Distributive | Let $\struct {T, +, \times}$ be an algebraic structure, and let $S$ be a set.
Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $+$ and $\times$.
Let $\times$ be left distributive over $+$:
:$\forall a, b, c \in S: a \times \paren {b + c} = \paren{a \times b} + \paren{a \times c}$
Then the point... | Let $f, g, h \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren{f \otimes \paren {g \oplus h} } } x
| r = \map f x \times \map {\paren{g \otimes h} } x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x \times \paren{ \map g x + \map h x}
| c = {{Defof|Pointwis... | Let $\struct {T, +, \times}$ be an [[Definition:Algebraic Structure with Two Operations|algebraic structure]], and let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus, \otimes}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $+$ and $\times$.
Let $\times$ be [[Definition:Left Distri... | Let $f, g, h \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren{f \otimes \paren {g \oplus h} } } x
| r = \map f x \times \map {\paren{g \otimes h} } x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \map f x \times \paren{ \map g x + \map h x}
| c = {{Defof|Point... | Structure Induced by Left Distributive Operation is Left Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Left_Distributive_Operation_is_Left_Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Left_Distributive_Operation_is_Left_Distributive | [
"Pointwise Operations",
"Distributive Operations"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Distributive Operation/Left",
"Definition:Pointwise Operation",
"Definition:Distributive Operation/Left",
"Definition:Pointwise Operation"
] | [
"Equality of Mappings",
"Definition:Distributive Operation/Left",
"Category:Pointwise Operations",
"Category:Distributive Operations"
] |
proofwiki-21497 | Structure Induced by Right Distributive Operation is Right Distributive | Let $\struct {T, +, \times}$ be an algebraic structure, and let $S$ be a set.
Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $+$ and $\times$.
Let $\times$ be right distributive over $+$:
:$\forall a, b, c \in S: \paren {b + c} \times a= \paren{b \times a} + \paren{c \times a}$
Then the point... | Let $f, g, h \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren{\paren {g \oplus h} \otimes f} } x
| r = \map {\paren{g \otimes h} } x \times \map f x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \paren{ \map g x + \map h x} \times \map f x
| c = {{Defof|Pointwise... | Let $\struct {T, +, \times}$ be an [[Definition:Algebraic Structure with Two Operations|algebraic structure]], and let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus, \otimes}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $+$ and $\times$.
Let $\times$ be [[Definition:Right Distr... | Let $f, g, h \in T^S$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren{\paren {g \oplus h} \otimes f} } x
| r = \map {\paren{g \otimes h} } x \times \map f x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = \paren{ \map g x + \map h x} \times \map f x
| c = {{Defof|Pointw... | Structure Induced by Right Distributive Operation is Right Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Right_Distributive_Operation_is_Right_Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Right_Distributive_Operation_is_Right_Distributive | [
"Pointwise Operations",
"Distributive Operations"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Distributive Operation/Right",
"Definition:Pointwise Operation",
"Definition:Distributive Operation/Left",
"Definition:Pointwise Operation"
] | [
"Equality of Mappings",
"Definition:Distributive Operation/Right",
"Category:Pointwise Operations",
"Category:Distributive Operations"
] |
proofwiki-21498 | Structure Induced by Distributive Operation is Distributive | Let $\struct {T, +, \times}$ be an algebraic structure, and let $S$ be a set.
Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $+$ and $\times$.
Let $\times$ be distributive over $+$:
:$\forall a, b, c \in S$:
::$a \times \paren {b + c} = \paren{a \times b} + \paren{a \times c}$
:and:
::$\pare... | By definition of distributive operation:
:$\times$ is left distributive over $+$
and
:$\times$ is right distributive over $+$
From Structure Induced by Left Distributive Operation is Left Distributive:
:$\otimes$ is left distributive over $\oplus$
From Structure Induced by Right Distributive Operation is Right Distribu... | Let $\struct {T, +, \times}$ be an [[Definition:Algebraic Structure with Two Operations|algebraic structure]], and let $S$ be a [[Definition:Set|set]].
Let $\struct {T^S, \oplus, \otimes}$ be the [[Definition:Induced Structure|structure on $T^S$ induced]] by $+$ and $\times$.
Let $\times$ be [[Definition:Distributiv... | By definition of [[Definition:Distributive Operation|distributive operation]]:
:$\times$ is [[Definition:Left Distributive Operation|left distributive]] over $+$
and
:$\times$ is [[Definition:Right Distributive Operation|right distributive]] over $+$
From [[Structure Induced by Left Distributive Operation is Left Dis... | Structure Induced by Distributive Operation is Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Distributive_Operation_is_Distributive | https://proofwiki.org/wiki/Structure_Induced_by_Distributive_Operation_is_Distributive | [
"Pointwise Operations",
"Distributive Operations"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Distributive Operation",
"Definition:Pointwise Operation",
"Definition:Distributive Operation",
"Definition:Pointwise Operation"
] | [
"Definition:Distributive Operation",
"Definition:Distributive Operation/Left",
"Definition:Distributive Operation/Right",
"Structure Induced by Left Distributive Operation is Left Distributive",
"Definition:Distributive Operation/Left",
"Structure Induced by Right Distributive Operation is Right Distribut... |
proofwiki-21499 | Phi in the Pentagon | Let $ABCDE$ be a regular pentagon.
The chords and sides of $ABCDE$ form three kinds of similar triangles.
* I: Isosceles, similar to $\triangle ACD$.
* II: Isosceles, similar to $\triangle ABE$.
The sides of the first type are in the ratio $\phi : 1$, where $\phi$ is the golden mean.
The sides of the second type are in... | {{hypothesis}} five sides of $ABCDE$ are equal.
Draw all five chords.
From symmetry, the chords are all equal.
$\leadsto$:
:$\triangle ACD$ is an isosceles triangle.
$\leadsto$:
:$\triangle BCD$ is an isosceles triangle
{{begin-eqn}}
{{eqn | l = \angle DBC
| r = \angle CDB = \alpha
| c = Isosceles Triangle ... | Let $ABCDE$ be a [[Definition:Regular Pentagon|regular pentagon]].
The [[Definition:Chord of Polygon|chords]] and [[Definition:Side of Polygon|sides]] of $ABCDE$ form three kinds of [[Definition:Similar Triangles|similar triangles]].
* I: [[Definition:Isosceles Triangle|Isosceles]], [[Definition:Similar Triangles|sim... | {{hypothesis}} five [[Definition:Side of Polygon|sides]] of $ABCDE$ are equal.
Draw all five [[Definition:Chord of Polygon|chords]].
From [[Definition:Symmetry|symmetry]], the [[Definition:Chord of Polygon|chords]] are all equal.
$\leadsto$:
:$\triangle ACD$ is an [[Definition:Isosceles Triangle|isosceles triangle]]... | Phi in the Pentagon | https://proofwiki.org/wiki/Phi_in_the_Pentagon | https://proofwiki.org/wiki/Phi_in_the_Pentagon | [
"Pentagons",
"Golden Mean"
] | [
"Definition:Pentagon/Regular",
"Definition:Polygon/Chord",
"Definition:Polygon/Side",
"Definition:Similar Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Similar Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Similar Triangles",
"Definition:Polygon/Side",
"... | [
"Definition:Polygon/Side",
"Definition:Polygon/Chord",
"Definition:Symmetry",
"Definition:Polygon/Chord",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles",
"Isosceles Triangle has Two Equal Angles",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Tr... |
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