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-22300 | Densely-Defined Linear Operator is Closable iff Adjoint is Densely-Defined | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a densely-defined linear operator.
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\map D T, T}$.
Then $\tuple {\map D T, T}$ is closable {{iff}} $\tuple {\map D {T^\ast}, T^\ast}$ is densely-define... | Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the Hilbert space direct sum of $\HH$ with itself.
From Equivalent Norms on Direct Product of Normed Vector Spaces, the inner product norm on $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ is equivalent to the direct produ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-Defined Linear Operator|a... | Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the [[Definition:Hilbert Space Direct Sum|Hilbert space direct sum]] of $\HH$ with itself.
From [[Equivalent Norms on Direct Product of Normed Vector Spaces]], the [[Definition:Inner Product Norm|inner product norm]] on $\struct {\HH \times \H... | Densely-Defined Linear Operator is Closable iff Adjoint is Densely-Defined | https://proofwiki.org/wiki/Densely-Defined_Linear_Operator_is_Closable_iff_Adjoint_is_Densely-Defined | https://proofwiki.org/wiki/Densely-Defined_Linear_Operator_is_Closable_iff_Adjoint_is_Densely-Defined | [
"Closable Densely-Defined Linear Operators",
"Adjoints (Densely-Defined Linear Operators)",
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Closu... | [
"Definition:Hilbert Space Direct Sum",
"Equivalent Norms on Direct Product of Normed Vector Spaces",
"Definition:Inner Product Norm",
"Definition:Equivalence of Norms",
"Definition:Direct Product Norm",
"Open Sets in Vector Spaces with Equivalent Norms Coincide",
"Square of V Operator on Hilbert Space",... |
proofwiki-22301 | Adjoint of Closure of Closable Densely-Defined Linear Operator | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a closable densely-defined linear operator with adjoint $\tuple {\map D {T^\ast}, T^\ast}$.
Let $\tuple {\map D {\overline T}, \overline T}$ be the closure of $T$ with adjoint $\tuple {\map D {\paren {\overline T}^\... | From Adjoint of Densely-Defined Linear Operator is Closed:
:$\tuple {\map D {T^\ast}, T^\ast}$ is closed.
From Closable Densely-Defined Operator is Closed iff Equal to Closure, we have:
:$\tuple {\map D {T^\ast}, T^\ast} = \tuple {\map D {\overline {T^\ast} }, \overline {T^\ast} }$
From Densely-Defined Linear Operator ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Closable Densely-Defined Linear Operator|closable densely-defined linear operator]] with [[Definition:Adjoint of Densely-Defined Linear Operator|adjoint]] $\tuple {\map D ... | From [[Adjoint of Densely-Defined Linear Operator is Closed]]:
:$\tuple {\map D {T^\ast}, T^\ast}$ is [[Definition:Closed Linear Transformation|closed]].
From [[Closable Densely-Defined Operator is Closed iff Equal to Closure]], we have:
:$\tuple {\map D {T^\ast}, T^\ast} = \tuple {\map D {\overline {T^\ast} }, \overl... | Adjoint of Closure of Closable Densely-Defined Linear Operator | https://proofwiki.org/wiki/Adjoint_of_Closure_of_Closable_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Adjoint_of_Closure_of_Closable_Densely-Defined_Linear_Operator | [
"Closable Densely-Defined Linear Operators",
"Adjoints (Densely-Defined Linear Operators)",
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Definition:Closure of Closable Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator"
] | [
"Adjoint of Densely-Defined Linear Operator is Closed",
"Definition:Closed Linear Transformation",
"Closable Densely-Defined Operator is Closed iff Equal to Closure",
"Densely-Defined Linear Operator is Closable iff Adjoint is Densely-Defined",
"Densely-Defined Linear Operator is Closable iff Adjoint is Den... |
proofwiki-22302 | Symmetric Densely-Defined Linear Operator is Closable | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a symmetric densely-defined linear operator.
Then $\tuple {\map D T, T}$ is closable. | Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\map D T, T}$.
From Adjoint of Densely-Defined Linear Operator is Closed, $\tuple {\map D {T^\ast}, T^\ast}$ is closed.
From Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator, $\tuple {\map D {T^\ast}, T^\ast}$ extends $\tuple {\map D... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Symmetric Densely-Defined Linear Operator|symmetric densely-defined linear operator]].
Then $\tuple {\map D T, T}$ is [[Definition:Closable Densely-Defined Linear Operat... | Let $\tuple {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-Defined Linear Operator|adjoint]] of $\tuple {\map D T, T}$.
From [[Adjoint of Densely-Defined Linear Operator is Closed]], $\tuple {\map D {T^\ast}, T^\ast}$ is [[Definition:Closed Linear Transformation|closed]].
From [[Adjoint of Symmetri... | Symmetric Densely-Defined Linear Operator is Closable | https://proofwiki.org/wiki/Symmetric_Densely-Defined_Linear_Operator_is_Closable | https://proofwiki.org/wiki/Symmetric_Densely-Defined_Linear_Operator_is_Closable | [
"Symmetric Densely-Defined Linear Operators",
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Closable Densely-Defined Linear Operator"
] | [
"Definition:Adjoint of Densely-Defined Linear Operator",
"Adjoint of Densely-Defined Linear Operator is Closed",
"Definition:Closed Linear Transformation",
"Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator",
"Definition:Extension of Mapping",
"Definition:Closable Densely-Defined Linea... |
proofwiki-22303 | Globally-Defined Symmetric Linear Operator is Bounded and Self-Adjoint | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\HH, T}$ be a symmetric linear operator.
Then $\tuple {\HH, T}$ is bounded and self-adjoint. | Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\HH, T}$.
From Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator, $\tuple {\map D {T^\ast}, T^\ast}$ extends $\tuple {\HH, T}$.
However, $\map D {T^\ast} \subseteq \HH$.
So we have $\map D {T^\ast} = \HH = \map D T$.
Hence $T^\ast ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\HH, T}$ be a [[Definition:Symmetric Densely-Defined Linear Operator|symmetric linear operator]].
Then $\tuple {\HH, T}$ is [[Definition:Bounded Linear Transformation|bounded]] and [[Definition:Self-Ad... | Let $\tuple {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-Defined Linear Operator|adjoint]] of $\tuple {\HH, T}$.
From [[Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator]], $\tuple {\map D {T^\ast}, T^\ast}$ [[Definition:Extension of Mapping|extends]] $\tuple {\HH, T}$.
Howe... | Globally-Defined Symmetric Linear Operator is Bounded and Self-Adjoint | https://proofwiki.org/wiki/Globally-Defined_Symmetric_Linear_Operator_is_Bounded_and_Self-Adjoint | https://proofwiki.org/wiki/Globally-Defined_Symmetric_Linear_Operator_is_Bounded_and_Self-Adjoint | [
"Symmetric Densely-Defined Linear Operators",
"Self-Adjoint Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Bounded Linear Transformation",
"Definition:Self-Adjoint Densely-Defined Linear Operator"
] | [
"Definition:Adjoint of Densely-Defined Linear Operator",
"Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator",
"Definition:Extension of Mapping",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Adjoint of Densely-Defined Linear Operator is Closed",
"Definition:Closed Linear ... |
proofwiki-22304 | Injective Self-Adjoint Densely-Defined Linear Operator has Everywhere Dense Image | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a self-adjoint densely defined linear operator that is injective.
Then $\Img T$ is everywhere dense in $\HH$. | From Kernel of Adjoint of Densely-Defined Linear Operator is Orthocomplement of Image, we have:
:$\map \ker {T^\ast} = \paren {\Img T}^\bot$
where $\tuple {\map D {T^\ast}, T^\ast}$ is the adjoint of $\tuple {\map D T, T}$.
Since $T$ is self-adjoint, we have $T^\ast = T$.
Hence $\map \ker T = \paren {\Img T}^\bot$.
S... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Self-Adjoint Densely-Defined Linear Operator|self-adjoint densely defined linear operator]] that is [[Definition:Injection|injective]].
Then $\Img T$ is [[Definition:Eve... | From [[Kernel of Adjoint of Densely-Defined Linear Operator is Orthocomplement of Image]], we have:
:$\map \ker {T^\ast} = \paren {\Img T}^\bot$
where $\tuple {\map D {T^\ast}, T^\ast}$ is the [[Definition:Adjoint of Densely-Defined Linear Operator|adjoint]] of $\tuple {\map D T, T}$.
Since $T$ is [[Definition:Self-A... | Injective Self-Adjoint Densely-Defined Linear Operator has Everywhere Dense Image | https://proofwiki.org/wiki/Injective_Self-Adjoint_Densely-Defined_Linear_Operator_has_Everywhere_Dense_Image | https://proofwiki.org/wiki/Injective_Self-Adjoint_Densely-Defined_Linear_Operator_has_Everywhere_Dense_Image | [
"Self-Adjoint Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Injection",
"Definition:Everywhere Dense"
] | [
"Kernel of Adjoint of Densely-Defined Linear Operator is Orthocomplement of Image",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Injection",
"Linear Subspace Dense iff Zero Orthocomplement",
"Definition:Everywhere Dense"
] |
proofwiki-22305 | Kernel of Closed Linear Transformation is Closed | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces.
Let $T : \map D T \to Y$ be a closed linear transformation.
Let $\ker T$ be the kernel of $T$.
Then $\ker T$ is closed. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\ker T$ converging to $x \in X$.
We have $T x_n = {\mathbf 0}_Y$ for each $n \in \N$.
Hence from Constant Sequence in Normed Vector Space Converges, we have $T x_n \to {\mathbf 0}_Y$ as $n \to \infty$.
From Sequential Characterization of Closed Linear Transfor... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $T : \map D T \to Y$ be a [[Definition:Closed Linear Transformation|closed linear transformation]].
Let $\ker T$ be the [[Definition:Kernel of Linear Transformation|kernel]] of $T$.
Th... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\ker T$ [[Definition:Convergent Sequence|converging to]] $x \in X$.
We have $T x_n = {\mathbf 0}_Y$ for each $n \in \N$.
Hence from [[Constant Sequence in Normed Vector Space Converges]], we have $T x_n \to {\mathbf 0}_Y$ as $n \to \i... | Kernel of Closed Linear Transformation is Closed | https://proofwiki.org/wiki/Kernel_of_Closed_Linear_Transformation_is_Closed | https://proofwiki.org/wiki/Kernel_of_Closed_Linear_Transformation_is_Closed | [
"Closed Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Closed Linear Transformation",
"Definition:Kernel of Linear Transformation",
"Definition:Closed Set"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence",
"Constant Sequence in Normed Vector Space Converges",
"Sequential Characterization of Closed Linear Transformation",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence",
"Definition:Closed Set",
"Category:Closed Linear Transformatio... |
proofwiki-22306 | Self-Adjoint Densely-Defined Linear Operator is Maximally Symmetric | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a self-adjoint densely-defined linear operator.
Then $\tuple {\map D T, T}$ is maximally symmetric. | Let $\tuple {\map D S, S}$ be a symmetric densely-defined linear operator extending $\tuple {\map D T, T}$.
Let $\tuple {\map D {S^\ast}, S^\ast}$ and $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoints of $\tuple {\map D S, S}$ and $\tuple {\map D T, T}$ respectively.
From Adjoint of Densely-Defined Linear Operator rev... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Self-Adjoint Densely-Defined Linear Operator|self-adjoint densely-defined linear operator]].
Then $\tuple {\map D T, T}$ is [[Definition:Maximally Symmetric Densely-Defi... | Let $\tuple {\map D S, S}$ be a [[Definition:Symmetric Densely-Defined Linear Operator|symmetric densely-defined linear operator]] [[Definition:Extension of Mapping|extending]] $\tuple {\map D T, T}$.
Let $\tuple {\map D {S^\ast}, S^\ast}$ and $\tuple {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-D... | Self-Adjoint Densely-Defined Linear Operator is Maximally Symmetric | https://proofwiki.org/wiki/Self-Adjoint_Densely-Defined_Linear_Operator_is_Maximally_Symmetric | https://proofwiki.org/wiki/Self-Adjoint_Densely-Defined_Linear_Operator_is_Maximally_Symmetric | [
"Maximally Symmetric Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Self-Adjoint Densely-Defined Linear Operator",
"Definition:Maximally Symmetric Densely-Defined Linear Operator"
] | [
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Extension of Mapping",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Adjoint of Densely-Defined Linear Operator reverses Extension of Mapping",
"Definition:Extension of Mapping",
"Definition:Self-Adjoint Densely-Defined Linear... |
proofwiki-22307 | Descartes's Solution to Quartic Equation | Let $P$ be the quartic equation:
:$a x^4 + b x^3 + c x^2 + d x + e = 0$
such that $a \ne 0$.
Then $P$ has solutions:
{{WIP|Work to be done to complete this}} | First we render the quartic into monic form:
:$x^4 + \dfrac b a x^3 + \dfrac c a x^2 + \dfrac d a x + \dfrac e a = 0$
Using a '''Tschirnhaus transformation''', $x = y - \dfrac b {4 a}$, we convert $P$ into the reduced quartic:
:$y^4 + p y^2 + q y + r = 0$
This is set identically equal to:
:$\paren {y^2 + \lambda y + m}... | Let $P$ be the [[Definition:Quartic Equation|quartic equation]]:
:$a x^4 + b x^3 + c x^2 + d x + e = 0$
such that $a \ne 0$.
Then $P$ has solutions:
{{WIP|Work to be done to complete this}} | First we render the [[Definition:Quartic Equation|quartic]] into [[Definition:Monic Polynomial|monic]] form:
:$x^4 + \dfrac b a x^3 + \dfrac c a x^2 + \dfrac d a x + \dfrac e a = 0$
Using a '''[[Definition:Tschirnhaus Transformation|Tschirnhaus transformation]]''', $x = y - \dfrac b {4 a}$, we convert $P$ into the [[D... | Descartes's Solution to Quartic Equation | https://proofwiki.org/wiki/Descartes's_Solution_to_Quartic_Equation | https://proofwiki.org/wiki/Descartes's_Solution_to_Quartic_Equation | [
"Descartes's Solution to Quartic Equation",
"Quartic Equations"
] | [
"Definition:Quartic Equation"
] | [
"Definition:Quartic Equation",
"Definition:Monic Polynomial",
"Definition:Tschirnhaus Transformation",
"Definition:Reduced Quartic",
"Definition:Identity (Equation)",
"Definition:Cubic Equation",
"Cardano's Formula",
"Definition:Quadratic Equation",
"Definition:Reduced Quartic"
] |
proofwiki-22308 | Characterization of Completely Prime Filter in Complete Lattice | Let $\struct{L, \vee, \wedge, \preceq}$ be a complete lattice.
Let $F \subseteq L$.
Then:
:$F$ is a completely prime filter
{{iff}}
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigwedge A \in F \iff \paren{\forall a \in A : a \in ... | === Necessary Condition ===
Let $F$ be a completely prime filter.
{{:Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition}} | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $F \subseteq L$.
Then:
:$F$ is a [[Definition:Completely Prime Filter|completely prime filter]]
{{iff}}
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ [[D... | === [[Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition|Necessary Condition]] ===
Let $F$ be a [[Definition:Completely Prime Filter|completely prime filter]].
{{:Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition}} | Characterization of Completely Prime Filter in Complete Lattice | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice | [
"Complete Lattices",
"Completely Prime Filters",
"Characterization of Completely Prime Filter in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Completely Prime Filter",
"Definition:Finite",
"Definition:Supremum of Set",
"Definition:Infimum of Set"
] | [
"Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition",
"Definition:Completely Prime Filter"
] |
proofwiki-22309 | Closure of Densely-Defined Bounded Linear Operator | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a densely-defined linear operator that is bounded.
Let $\widetilde T : \HH \to \HH$ be the bounded linear operator extending $T$, given by Bounded Linear Transformation to Banach Space has Unique Extension to Closur... | From Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph and Closable Densely-Defined Linear Operator has Smallest Closed Extension, to show that the closure of $\tuple {\map D T, T}$ is $\tuple {\HH, \widetilde T}$, it is enough to show that:
:$\map \cl {\map \GG T} = \map \GG {\... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]] that is [[Definition:Bounded Linear Operator|bounded]].
Let $\widetilde T : \HH \to \HH$ be the [[Defin... | From [[Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph]] and [[Closable Densely-Defined Linear Operator has Smallest Closed Extension]], to show that the [[Definition:Closure of Closable Densely-Defined Linear Operator|closure]] of $\tuple {\map D T, T}$ is $\tuple {\HH, \wide... | Closure of Densely-Defined Bounded Linear Operator | https://proofwiki.org/wiki/Closure_of_Densely-Defined_Bounded_Linear_Operator | https://proofwiki.org/wiki/Closure_of_Densely-Defined_Bounded_Linear_Operator | [
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Extension of Mapping",
"Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain",
"Definition:Closable Densely... | [
"Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph",
"Closable Densely-Defined Linear Operator has Smallest Closed Extension",
"Definition:Closure of Closable Densely-Defined Linear Operator",
"Definition:Closure (Topology)",
"Definition:Direct Product Norm",
"Defi... |
proofwiki-22310 | Sum of Closable Densely-Defined Linear Operator and Densely-Defined Bounded Linear Operator is Closable | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a closable densely-defined linear operator with closure $\tuple {\map D {\overline T}, \overline T}$.
Let $\tuple {\map D S, S}$ be a densely-defined bounded linear operator with closure $\tuple {\HH, \overline S}$ s... | From Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph and Closable Densely-Defined Linear Operator has Smallest Closed Extension, it is enough to show that:
:$\map \cl {\map \GG {T + S} } = \map \GG {\overline T + \overline S}$
where $\GG$ denotes the graph.
Let $\tuple {x, y} ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Closable Densely-Defined Linear Operator|closable densely-defined linear operator]] with [[Definition:Closure of Closable Densely-Defined Linear Operator|closure]] $\tuple... | From [[Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph]] and [[Closable Densely-Defined Linear Operator has Smallest Closed Extension]], it is enough to show that:
:$\map \cl {\map \GG {T + S} } = \map \GG {\overline T + \overline S}$
where $\GG$ denotes the [[Definition:Graph... | Sum of Closable Densely-Defined Linear Operator and Densely-Defined Bounded Linear Operator is Closable | https://proofwiki.org/wiki/Sum_of_Closable_Densely-Defined_Linear_Operator_and_Densely-Defined_Bounded_Linear_Operator_is_Closable | https://proofwiki.org/wiki/Sum_of_Closable_Densely-Defined_Linear_Operator_and_Densely-Defined_Bounded_Linear_Operator_is_Closable | [
"Sum of Closable Densely-Defined Linear Operator and Densely-Defined Bounded Linear Operator is Closable",
"Closable Densely-Defined Linear Operators",
"Sum of Closable Densely-Defined Linear Operator and Densely-Defined Bounded Linear Operator is Closable"
] | [
"Definition:Hilbert Space",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Closure of Closable Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Closure of Closable Densely-Defined Linear Operator",
"Defi... | [
"Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph",
"Closable Densely-Defined Linear Operator has Smallest Closed Extension",
"Definition:Graph of Mapping",
"Definition:Closure (Topology)",
"Definition:Sequence",
"Convergence in Direct Product Norm",
"Definition... |
proofwiki-22311 | Measure is Monotone/Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B \HH$ be the space of bounded linear transformations on $\HH$.
Let $\le_{\map B \HH}$ be the canonical preordering of $\map B \HH$.
Let $\EE : \map ... | Write:
:$F = E \cup \paren {F \setminus E}$
From $(4)$ in the definition of a resolution of the identity, we have:
:$\map \EE F = \map \EE E + \map \EE {F \setminus E}$
From Bounds on Projection in Unital C*-Algebra, we have:
:${\mathbf 0}_{\map B \HH} \le_{\map B \HH} \map \EE {F \setminus E}$
From $(1)$ in the defi... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B \HH$ be the [[Definition:Space of Bounded Linea... | Write:
:$F = E \cup \paren {F \setminus E}$
From $(4)$ in the definition of a [[Definition:Resolution of the Identity|resolution of the identity]], we have:
:$\map \EE F = \map \EE E + \map \EE {F \setminus E}$
From [[Bounds on Projection in Unital C*-Algebra]], we have:
:${\mathbf 0}_{\map B \HH} \le_{\map B \HH} \... | Measure is Monotone/Resolution of the Identity | https://proofwiki.org/wiki/Measure_is_Monotone/Resolution_of_the_Identity | https://proofwiki.org/wiki/Measure_is_Monotone/Resolution_of_the_Identity | [
"Resolutions of the Identity",
"Measure is Monotone"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Resolution of the Identity"
] | [
"Definition:Resolution of the Identity",
"Bounds on Projection in Unital C*-Algebra",
"Definition:Preordered Vector Space",
"Category:Resolutions of the Identity",
"Category:Measure is Monotone"
] |
proofwiki-22312 | Bounds on Projection in Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $p$ be a projection on $A$.
Then we have:
:${\mathbf 0}_A \le_A p \le_A {\mathbf 1}_A$ | Since $p$ is a projection, it is Hermitian.
From Spectrum of Projection in *-Algebra: Corollary, we have:
:$\map {\sigma_A} p \subseteq \set {0, 1}$
From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, we have:
:${\mathbf 0}_A \le_A p \le_A {\mathbf 1}_A$
{{qed}}
Category:Projections (*... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $p$ be a [[Definition:Projection (*-Algebras)|projection]] on $A$.
... | Since $p$ is a [[Definition:Projection (*-Algebras)|projection]], it is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
From [[Spectrum of Projection in *-Algebra/Corollary|Spectrum of Projection in *-Algebra: Corollary]], we have:
:$\map {\sigma_A} p \subseteq \set {0, 1}$
From [[Bounds on Hermitian Element... | Bounds on Projection in Unital C*-Algebra | https://proofwiki.org/wiki/Bounds_on_Projection_in_Unital_C*-Algebra | https://proofwiki.org/wiki/Bounds_on_Projection_in_Unital_C*-Algebra | [
"Projections (*-Algebras)",
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Projection (*-Algebras)"
] | [
"Definition:Projection (*-Algebras)",
"Definition:Hermitian Element of *-Algebra",
"Spectrum of Projection in *-Algebra/Corollary",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Category:Projections (*-Algebras)",
"Category:Canonical Preorderings of C*-Algebras"
] |
proofwiki-22313 | Characterization of Canonical Preordering of Projections on Hilbert Space | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B \HH$ be the space of bounded linear operators on $\HH$ understood as a $\text C^\ast$-algebra.
Let $\le_{\map B \HH}$ be the canonical preordering of $\map B \HH$.
Let $P, Q$ be Hilbert space projections on $\HH$.
{{TFAE}}
:$(1) \qua... | We first note that from Characterization of Projections, $P$ is Hermitian, and that $P, Q$ is are orthogonal projections onto $\Img P$ and $\Img Q$ respectively. | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B \HH$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear operators]] on $\HH$ understood as a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_{\map B \HH}$ be ... | We first note that from [[Characterization of Projections]], $P$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]], and that $P, Q$ is are [[Definition:Orthogonal Projection|orthogonal projections]] onto $\Img P$ and $\Img Q$ respectively. | Characterization of Canonical Preordering of Projections on Hilbert Space | https://proofwiki.org/wiki/Characterization_of_Canonical_Preordering_of_Projections_on_Hilbert_Space | https://proofwiki.org/wiki/Characterization_of_Canonical_Preordering_of_Projections_on_Hilbert_Space | [
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Projection (Hilbert Spaces)",
"Definition:Projection (Hilbert Spaces)"
] | [
"Characterization of Projections",
"Definition:Hermitian Element of *-Algebra",
"Definition:Orthogonal Projection",
"Characterization of Projections",
"Definition:Hermitian Element of *-Algebra",
"Definition:Hermitian Element of *-Algebra"
] |
proofwiki-22314 | Everywhere Dense Subset of Countable Hilbert Space Direct Sum in terms of Everywhere Dense Subsets of Summands | Let $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$ be a sequence of Hilbert spaces over $\C$.
For each $n \in \N$, let $\family {e^{(n)}_\alpha}_{\alpha \in \map J n}$ be an everywhere dense subset of $\HH_n$.
Let $\tuple {\HH, \innerprod \cdot \cdot}$ be the Hilbert space direct sum of $\fami... | Let $f \in \HH$ and $\epsilon > 0$.
We need to show that there exists $g \in \SS$ with $\norm {f - g} < \epsilon$.
Then by definition of the Hilbert space direct sum and Net Convergence Equivalent to Absolute Convergence:
:$\ds \sum_{n \mathop = 1}^\infty \norm {\map f n}_{\HH_n}^2 < \infty$
Hence there exists $N \in ... | Let $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Hilbert Space|Hilbert spaces]] over $\C$.
For each $n \in \N$, let $\family {e^{(n)}_\alpha}_{\alpha \in \map J n}$ be an [[Definition:Everywhere Dense|everywhere dense subset]] of $\HH_n$... | Let $f \in \HH$ and $\epsilon > 0$.
We need to show that there exists $g \in \SS$ with $\norm {f - g} < \epsilon$.
Then by definition of the [[Definition:Hilbert Space Direct Sum|Hilbert space direct sum]] and [[Net Convergence Equivalent to Absolute Convergence]]:
:$\ds \sum_{n \mathop = 1}^\infty \norm {\map f n}_... | Everywhere Dense Subset of Countable Hilbert Space Direct Sum in terms of Everywhere Dense Subsets of Summands | https://proofwiki.org/wiki/Everywhere_Dense_Subset_of_Countable_Hilbert_Space_Direct_Sum_in_terms_of_Everywhere_Dense_Subsets_of_Summands | https://proofwiki.org/wiki/Everywhere_Dense_Subset_of_Countable_Hilbert_Space_Direct_Sum_in_terms_of_Everywhere_Dense_Subsets_of_Summands | [
"Hilbert Space Direct Sums"
] | [
"Definition:Sequence",
"Definition:Hilbert Space",
"Definition:Everywhere Dense",
"Definition:Hilbert Space Direct Sum",
"Definition:Everywhere Dense"
] | [
"Definition:Hilbert Space Direct Sum",
"Net Convergence Equivalent to Absolute Convergence",
"Category:Hilbert Space Direct Sums"
] |
proofwiki-22315 | Direct Sum of Densely-Defined Linear Operators is Densely-Defined Linear Operator | Let $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$ be a sequence of Hilbert spaces over $\C$.
Let $\tuple {\HH, \innerprod \cdot \cdot}$ be the Hilbert space direct sum of $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$.
For each $n \in \N$, let $\tuple {\map D {T_n}, ... | Let $\SS$ be the set of $f \in \HH$ such that:
:there exists $N \in \N$ such that:
::$\map f n = {\mathbf 0}_{\HH_n}$ for $n > N$.
For $\phi = \sequence {\phi_n}_{n \mathop \in \N} \in \SS$ and $N \in \N$ picked as a witness to the above condition, we have:
:$\ds \sum_{n \mathop = 1}^\infty \norm {T_n \phi_n}_{\HH_n}^2... | Let $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Hilbert Space|Hilbert spaces]] over $\C$.
Let $\tuple {\HH, \innerprod \cdot \cdot}$ be the [[Definition:Hilbert Space Direct Sum|Hilbert space direct sum]] of $\family {\tuple {\HH_n, \in... | Let $\SS$ be the [[Definition:Set|set]] of $f \in \HH$ such that:
:there exists $N \in \N$ such that:
::$\map f n = {\mathbf 0}_{\HH_n}$ for $n > N$.
For $\phi = \sequence {\phi_n}_{n \mathop \in \N} \in \SS$ and $N \in \N$ picked as a witness to the above condition, we have:
:$\ds \sum_{n \mathop = 1}^\infty \norm {T... | Direct Sum of Densely-Defined Linear Operators is Densely-Defined Linear Operator | https://proofwiki.org/wiki/Direct_Sum_of_Densely-Defined_Linear_Operators_is_Densely-Defined_Linear_Operator | https://proofwiki.org/wiki/Direct_Sum_of_Densely-Defined_Linear_Operators_is_Densely-Defined_Linear_Operator | [
"Direct Sums of Densely-Defined Linear Operators"
] | [
"Definition:Sequence",
"Definition:Hilbert Space",
"Definition:Hilbert Space Direct Sum",
"Definition:Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator"
] | [
"Definition:Set",
"Everywhere Dense Subset of Countable Hilbert Space Direct Sum in terms of Everywhere Dense Subsets of Summands",
"Definition:Everywhere Dense",
"Set Closure Preserves Set Inclusion",
"Definition:Everywhere Dense",
"Definition:Linear Transformation",
"Definition:Linear Transformation",... |
proofwiki-22316 | Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition | Let $\struct{L, \vee, \wedge, \preceq}$ be a complete lattice.
Let $F \subseteq L$ be a completely prime filter.
Then:
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigwedge A \in F \iff \paren{\forall a \in A : a \in F}$
where:
:*... | ==== $F$ satisfies Statement $(1)$ ====
Let $A \subseteq L$.
Let $\bigvee A \in F$
By definition of completely prime filter:
:$A \cap F \ne \O$
Let $x \in A \cap F$
By definition of set intersection:
:$x \in A$ and $x \in F$
Let $x \in A : x \in F$
By definition of supremum:
:$x \preceq \bigvee A$
By definition of filt... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $F \subseteq L$ be a [[Definition:Completely Prime Filter|completely prime filter]].
Then:
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ [[Definition:Fin... | ==== $F$ satisfies Statement $(1)$ ====
Let $A \subseteq L$.
Let $\bigvee A \in F$
By definition of [[Definition:Completely Prime Filter|completely prime filter]]:
:$A \cap F \ne \O$
Let $x \in A \cap F$
By definition of [[Definition:Set Intersection|set intersection]]:
:$x \in A$ and $x \in F$
Let $x \in A : x ... | Characterization of Completely Prime Filter in Complete Lattice/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice/Necessary_Condition | [
"Characterization of Completely Prime Filter in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Completely Prime Filter",
"Definition:Finite",
"Definition:Supremum of Set",
"Definition:Infimum of Set"
] | [
"Definition:Completely Prime Filter",
"Definition:Set Intersection",
"Definition:Supremum of Set",
"Definition:Filter",
"Definition:Finite Subset",
"Definition:Infimum of Set",
"Definition:Filter",
"Definition:Filter",
"Definition:Sublattice",
"Existence of Non-Empty Finite Infima in Meet Semilatt... |
proofwiki-22317 | Characterization of Completely Prime Filter in Complete Lattice/Sufficient Condition | Let $\struct{L, \vee, \wedge, \preceq}$ be a complete lattice.
Let $F \subseteq L$ satisfy:
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigwedge A \in F \iff \paren{\forall a \in A : a \in F}$
where:
:* $\bigvee A$ denotes the su... | ==== $F$ is a Proper Subset ====
From Supremum of Empty Set is Smallest Element:
:$\bot = \bigvee \O$
By definition of empty set:
:$\forall x \in F : x \notin \O$
By the contrapositive statement of $(1)$:
:$\bot = \bigvee \O \notin F$
It follows that $F \neq L$
{{qed|lemma}}
==== $F$ is not Empty ====
From Infimum of E... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $F \subseteq L$ satisfy:
:$(1)\quad\forall A \subseteq L : \bigvee A \in F \iff \paren{\exists a \in A : a \in F}$
:$(2)\quad\forall $ [[Definition:Finite|finite]] $A \subseteq L : \bigwedge A \in F \iff \paren{\forall ... | ==== $F$ is a Proper Subset ====
From [[Supremum of Empty Set is Smallest Element]]:
:$\bot = \bigvee \O$
By definition of [[Definition:Empty Set|empty set]]:
:$\forall x \in F : x \notin \O$
By the [[Definition:Contrapositive Statement|contrapositive statement]] of $(1)$:
:$\bot = \bigvee \O \notin F$
It follows t... | Characterization of Completely Prime Filter in Complete Lattice/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Filter_in_Complete_Lattice/Sufficient_Condition | [
"Characterization of Completely Prime Filter in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Finite",
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Completely Prime Filter"
] | [
"Supremum of Empty Set is Smallest Element",
"Definition:Empty Set",
"Definition:Contrapositive Statement",
"Infimum of Empty Set is Greatest Element",
"Definition:Empty Set",
"Definition:Vacuous",
"Axiom:Filter Axioms",
"Meet Precedes Operands",
"Axiom:Filter Axioms",
"Successor is Supremum",
"... |
proofwiki-22318 | Eigenstates of Hamiltonian have Well Defined Symmetry only if Potential is Even | Given a certain Hamiltonian describing a quantum system, its eigenstates will have a well-defined symmetry in space (even or odd) only if the potential said system is subjected to is an even function of space | Given some Hamiltonian $H$, we can find its eigenfunctions in order to determine the eigenstates of the system, thus forming an orthogonal (or orthonormal, if normalized) base of the corresponding Hilbert space. If these eigenfunctions are to have a well-defined symmetry, then they must also be eigenfunctions of the pa... | Given a certain Hamiltonian describing a quantum system, its eigenstates will have a well-defined symmetry in space (even or odd) only if the potential said system is subjected to is an even function of space | Given some Hamiltonian $H$, we can find its eigenfunctions in order to determine the eigenstates of the system, thus forming an orthogonal (or orthonormal, if normalized) base of the corresponding Hilbert space. If these eigenfunctions are to have a well-defined symmetry, then they must also be eigenfunctions of the pa... | Eigenstates of Hamiltonian have Well Defined Symmetry only if Potential is Even | https://proofwiki.org/wiki/Eigenstates_of_Hamiltonian_have_Well_Defined_Symmetry_only_if_Potential_is_Even | https://proofwiki.org/wiki/Eigenstates_of_Hamiltonian_have_Well_Defined_Symmetry_only_if_Potential_is_Even | [] | [] | [] |
proofwiki-22319 | Complex-Valued Function is Measurable iff Real and Imaginary Part Measurable | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $\struct {\R, \map \BB \R}$ be the real numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $f : X \to \C$ be a function.
Let $\... | === Necessary Condition ===
Suppose that $f$ is $\struct {X, \Sigma}$/$\struct {\C, \map \BB \C}$-measurable.
From Real and Imaginary Part Projections are Continuous and Continuous Mapping is Measurable, the real part and imaginary part are $\map \BB \C/\map \BB \R$-measurable as mappings $\C \to \R$.
Hence from Compos... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\struct {\C, \map \BB \C}$ be the [[Definition:Complex Number|complex numbers]] made into a [[Definition:Measurable Space|measurable space]] with its [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]].
Let $\struct {\R, \map ... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Measurable Mapping|$\struct {X, \Sigma}$/$\struct {\C, \map \BB \C}$-measurable]].
From [[Real and Imaginary Part Projections are Continuous]] and [[Continuous Mapping is Measurable]], the [[Definition:Real Part|real part]] and [[Definition:Imaginary Part|i... | Complex-Valued Function is Measurable iff Real and Imaginary Part Measurable | https://proofwiki.org/wiki/Complex-Valued_Function_is_Measurable_iff_Real_and_Imaginary_Part_Measurable | https://proofwiki.org/wiki/Complex-Valued_Function_is_Measurable_iff_Real_and_Imaginary_Part_Measurable | [
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Complex Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Real Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Function",
"Definition:Complex Number/Real Part",
"Definition:Complex ... | [
"Definition:Measurable Mapping",
"Real and Imaginary Part Projections are Continuous",
"Continuous Mapping is Measurable",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Measurable Mapping",
"Definition:Mapping",
"Composition of Measurable Mappings is Me... |
proofwiki-22320 | Measurability in Trace Sigma-Algebra | Let $\struct {A, \Sigma_A}$ and $\struct {C, \Sigma_C}$ be measurable spaces.
Let $B \subseteq C$.
Let $\Sigma_B$ be the trace $\sigma$-algebra on $B$ induced by $\Sigma_C$.
Let $f : A \to B$ be $\Sigma_A/\Sigma_B$-measurable.
Then $f$ is $\Sigma_A/\Sigma_C$-measurable. | Let $S \in \Sigma_C$.
By Image is Subset of Codomain/Corollary 2, we have:
:$f \sqbrk A \subseteq B$
we have:
:$f^{-1} \sqbrk S = f^{-1} \sqbrk {S \cap B}$
From the definition of the trace $\sigma$-algebra, we have:
:$S \cap B \in \Sigma_B$
Since $f$ is $\Sigma_A/\Sigma_B$-measurable, we have:
:$f^{-1} \sqbrk {S \cap B... | Let $\struct {A, \Sigma_A}$ and $\struct {C, \Sigma_C}$ be [[Definition:Measurable Space|measurable spaces]].
Let $B \subseteq C$.
Let $\Sigma_B$ be the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra]] on $B$ induced by $\Sigma_C$.
Let $f : A \to B$ be [[Definition:Measurable Mapping|$\Sigma_A/\Sigma_B$-m... | Let $S \in \Sigma_C$.
By [[Image is Subset of Codomain/Corollary 2]], we have:
:$f \sqbrk A \subseteq B$
we have:
:$f^{-1} \sqbrk S = f^{-1} \sqbrk {S \cap B}$
From the definition of the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra]], we have:
:$S \cap B \in \Sigma_B$
Since $f$ is [[Definition:Measurable... | Measurability in Trace Sigma-Algebra | https://proofwiki.org/wiki/Measurability_in_Trace_Sigma-Algebra | https://proofwiki.org/wiki/Measurability_in_Trace_Sigma-Algebra | [
"Measurable Functions",
"Trace Sigma-Algebras"
] | [
"Definition:Measurable Space",
"Definition:Trace Sigma-Algebra",
"Definition:Measurable Mapping",
"Definition:Measurable Mapping"
] | [
"Image is Subset of Codomain/Corollary 2",
"Definition:Trace Sigma-Algebra",
"Definition:Measurable Mapping",
"Definition:Measurable Mapping",
"Category:Measurable Functions",
"Category:Trace Sigma-Algebras"
] |
proofwiki-22321 | Linear Combination of Measurable Functions valued in Topological Vector Space is Measurable | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field be a topological field.
Let $\struct {Y, \tau}$ be a topological vector space over $K$.
Let $\map \BB Y$ be the Borel $\sigma$-algebra of $\struct {Y, \tau}$.
Let $f : X \to Y$ and $g : X \to Y$ be $\map \BB ... | Define $m_\lambda : Y \to Y$ by:
:$\map {m_\lambda} y = \lambda y$
for each $y \in Y$.
By the definition of a topological vector space, $m_\lambda$ is continuous.
Hence from Continuous Mapping is Measurable, $m_\lambda$ is $\map \BB Y/\map \BB Y$-measurable.
Equip $Y^2$ with the product topology.
Define also $s : Y^2 ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\struct {K, +_K, \circ_K, \tau_K}$ be a [[Definition:Topological Field|topological field]] be a [[Definition:Topological Field|topological field]].
Let $\struct {Y, \tau}$ be a [[Definition:Topological Vector Space|topological vecto... | Define $m_\lambda : Y \to Y$ by:
:$\map {m_\lambda} y = \lambda y$
for each $y \in Y$.
By the definition of a [[Definition:Topological Vector Space|topological vector space]], $m_\lambda$ is [[Definition:Continuous Mapping|continuous]].
Hence from [[Continuous Mapping is Measurable]], $m_\lambda$ is [[Definition:Mea... | Linear Combination of Measurable Functions valued in Topological Vector Space is Measurable | https://proofwiki.org/wiki/Linear_Combination_of_Measurable_Functions_valued_in_Topological_Vector_Space_is_Measurable | https://proofwiki.org/wiki/Linear_Combination_of_Measurable_Functions_valued_in_Topological_Vector_Space_is_Measurable | [
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Topological Field",
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Borel Sigma-Algebra",
"Definition:Measurable Mapping",
"Definition:Measurable Mapping"
] | [
"Definition:Topological Vector Space",
"Definition:Continuous Mapping",
"Continuous Mapping is Measurable",
"Definition:Measurable Mapping",
"Definition:Product Topology",
"Definition:Topological Vector Space",
"Definition:Continuous Mapping",
"Continuous Mapping is Measurable",
"Definition:Measurab... |
proofwiki-22322 | Content of Subdivided Rectangle | Let $R$ be a closed $n$-rectangle.
Let $P$ be a subdivision of $R$.
Then:
:$\ds \map V R = \sum_{r \mathop \in P^*} \map V r$
where:
:$\map V R$ denotes the content of the rectangle $R$
:$P^*$ denotes the set of subrectangles of $P$. | We will proceed by induction on $n$. | Let $R$ be a [[Definition:Closed Rectangle|closed $n$-rectangle]].
Let $P$ be a [[Definition:Finite Subdivision of Rectangle|subdivision]] of $R$.
Then:
:$\ds \map V R = \sum_{r \mathop \in P^*} \map V r$
where:
:$\map V R$ denotes the [[Definition:Content of Rectangle|content of the rectangle $R$]]
:$P^*$ denotes t... | We will proceed by [[Definition:Mathematical Induction|induction]] on $n$. | Content of Subdivided Rectangle | https://proofwiki.org/wiki/Content_of_Subdivided_Rectangle | https://proofwiki.org/wiki/Content_of_Subdivided_Rectangle | [
"Jordan Content"
] | [
"Definition:Closed Rectangle",
"Definition:Subdivision of Interval/Rectangle",
"Definition:Content of Rectangle",
"Definition:Set",
"Definition:Subdivision of Interval/Rectangle/Subrectangle"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-22323 | Homogeneity of Space implies Conservation of Linear Momentum | Because space is homogeneous in an inertial frame of reference, linear momentum is conserved if the system is closed. | In inertial frames of reference, space is homogeneous. Therefore, working exclusively with inertial frames of reference, in closed systems (those which do not interact in any ways with external agents and are therefore causally disconnected from anything external), we can make a translation of every particle in said sy... | Because space is homogeneous in an [[Definition:Inertial Frame of Reference|inertial frame of reference]], [[Definition:Linear Momentum|linear momentum]] is conserved if the system is closed. | In inertial frames of reference, space is homogeneous. Therefore, working exclusively with inertial frames of reference, in closed systems (those which do not interact in any ways with external agents and are therefore causally disconnected from anything external), we can make a translation of every particle in said sy... | Homogeneity of Space implies Conservation of Linear Momentum | https://proofwiki.org/wiki/Homogeneity_of_Space_implies_Conservation_of_Linear_Momentum | https://proofwiki.org/wiki/Homogeneity_of_Space_implies_Conservation_of_Linear_Momentum | [] | [
"Definition:Inertial Frame of Reference",
"Definition:Linear Momentum"
] | [
"Definition:Euler-Lagrange Equation"
] |
proofwiki-22324 | Decimal Expansion of Number is Terminating or Periodic iff Rational | Let $x$ be a real number.
Then:
:the decimal expansion of $x$ is either terminating or recurring
{{iff}}
:$x$ is a rational number. | === Necessary Condition ===
Let $x$ be a rational number.
Let $x$ be expressed in the form $\dfrac a b$ where $a \in \Z$ and $b \in \Z_{>0}$.
By the division algorithm:
:$a = b q + r$
for some integers $q, r$ and $0 \le r < b$.
It follows that if $r = 0$ the decimal terminates.
Suppose $r \ne 0$.
We have that $r$ is an... | Let $x$ be a [[Definition:Real Number|real number]].
Then:
:the [[Definition:Decimal Expansion|decimal expansion]] of $x$ is either [[Definition:Terminating Decimal|terminating]] or [[Definition:Recurring Decimal|recurring]]
{{iff}}
:$x$ is a [[Definition:Rational Number|rational number]]. | === Necessary Condition ===
Let $x$ be a [[Definition:Rational Number|rational number]].
Let $x$ be expressed in the form $\dfrac a b$ where $a \in \Z$ and $b \in \Z_{>0}$.
By the division algorithm:
:$a = b q + r$
for some integers $q, r$ and $0 \le r < b$.
It follows that if $r = 0$ the [[Definition:Terminating ... | Decimal Expansion of Number is Terminating or Periodic iff Rational | https://proofwiki.org/wiki/Decimal_Expansion_of_Number_is_Terminating_or_Periodic_iff_Rational | https://proofwiki.org/wiki/Decimal_Expansion_of_Number_is_Terminating_or_Periodic_iff_Rational | [
"Rational Numbers",
"Terminating Decimals",
"Recurring Decimals"
] | [
"Definition:Real Number",
"Definition:Decimal Expansion",
"Definition:Terminating Decimal",
"Definition:Recurring Decimal",
"Definition:Rational Number"
] | [
"Definition:Rational Number",
"Definition:Terminating Decimal",
"Definition:Element",
"Definition:Finite Set",
"Definition:Integer",
"Dirichlet's Box Principle/Corollary",
"Definition:Decimal Expansion",
"Definition:Rational Number",
"Definition:Terminating Decimal",
"Definition:Recurring Decimal"... |
proofwiki-22325 | Borel-Carathéodory Lemma/Lemma | :$\ds \forall n \in \Z_{\ge 1} : \quad \frac {\cmod {\map {f^{\paren n} } 0} }{ n! } \le \frac {2 M} {R^n}$ | By Cauchy Integral Theorem:
:$\ds \forall k \in \Z_{\ge 0} : \quad \oint_{\partial D} z^{k-1} \map f z \rd z = 0$
Parametrizing $\partial D$ by $R e^{2 \pi i t}$:
:$\ds \forall k \in \Z_{\ge 0} : \quad \int _0^1 e^{2\pi i k t} \map f {R e^{2 \pi ikt} } \rd t = 0$
On the other hand:
{{begin-eqn}}
{{eqn | l = \map {f^{\p... | :$\ds \forall n \in \Z_{\ge 1} : \quad \frac {\cmod {\map {f^{\paren n} } 0} }{ n! } \le \frac {2 M} {R^n}$ | By [[Cauchy Integral Theorem]]:
:$\ds \forall k \in \Z_{\ge 0} : \quad \oint_{\partial D} z^{k-1} \map f z \rd z = 0$
Parametrizing $\partial D$ by $R e^{2 \pi i t}$:
:$\ds \forall k \in \Z_{\ge 0} : \quad \int _0^1 e^{2\pi i k t} \map f {R e^{2 \pi ikt} } \rd t = 0$
On the other hand:
{{begin-eqn}}
{{eqn | l = \map {... | Borel-Carathéodory Lemma/Lemma | https://proofwiki.org/wiki/Borel-Carathéodory_Lemma/Lemma | https://proofwiki.org/wiki/Borel-Carathéodory_Lemma/Lemma | [
"Complex Analysis"
] | [] | [
"Cauchy-Goursat Theorem",
"Cauchy's Integral Formula/General Result",
"Category:Complex Analysis"
] |
proofwiki-22326 | Content of Finite Set of Rectangles not Less than Covered Subrectangles | Let $S$ be a finite set of closed $n$-rectangles.
Let $\RR$ be a closed $n$-rectangle that contains every rectangle in $S$.
Let:
:$P = \tuple {P_1, \dotsc, P_n}$
be a subdivision of $\RR$ such that for every:
:$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i} \in S$
we have that:
:$a_i, b_i \in P_i$
That is, the ... | {{WIP}}
apyo4f6noaqs5oem3dn8fbmgqddd3u3 | Let $S$ be a [[Definition:Finite Set|finite set]] of [[Definition:Closed Rectangle|closed $n$-rectangles]].
Let $\RR$ be a [[Definition:Closed Rectangle|closed $n$-rectangle]] that [[Definition:Contain|contains]] every [[Definition:Closed Rectangle|rectangle]] in $S$.
Let:
:$P = \tuple {P_1, \dotsc, P_n}$
be a [[Defi... | {{WIP}}
apyo4f6noaqs5oem3dn8fbmgqddd3u3 | Content of Finite Set of Rectangles not Less than Covered Subrectangles | https://proofwiki.org/wiki/Content_of_Finite_Set_of_Rectangles_not_Less_than_Covered_Subrectangles | https://proofwiki.org/wiki/Content_of_Finite_Set_of_Rectangles_not_Less_than_Covered_Subrectangles | [] | [
"Definition:Finite Set",
"Definition:Closed Rectangle",
"Definition:Closed Rectangle",
"Definition:Subset",
"Definition:Closed Rectangle",
"Definition:Subdivision of Interval/Rectangle",
"Definition:Interval/Ordered Set/Endpoint",
"Definition:Element",
"Definition:Subdivision of Interval/Finite",
... | [] |
proofwiki-22327 | Meet Irreducible Element Induced by Completely Prime Filter | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\top$ denote the greatest element of $L$.
Let $p$ be a completely prime filter of $L$.
Let:
:$b = \bigvee \set{a \in L : a \notin p}$
where:
:$\bigvee \set{a \in L : a \notin p}$ denotes the supremum of the set $\set{a \in L : a \notin p}$
Then:
:$b$ is a meet ir... | Let:
:$x, y \in S : x \wedge y = b$
By definition of infimum:
:$b \preceq x$
:$b \preceq y$
By the contrapositive statement of the definition of completely prime filter:
:$b \notin p$
By the contrapositive statement of filter axiom $(2)$:
:either $x \notin p$ or $y \notin p$
By definition of supremum:
:either $x \prec... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\top$ denote the [[Definition:Greatest Element|greatest element]] of $L$.
Let $p$ be a [[Definition:Completely Prime Filter|completely prime filter]] of $L$.
Let:
:$b = \bigvee \set{a \in L : a \notin p}$
where:
:$\bigvee... | Let:
:$x, y \in S : x \wedge y = b$
By definition of [[Definition:Infimum of Set|infimum]]:
:$b \preceq x$
:$b \preceq y$
By the [[Definition:Contrapositive Statement|contrapositive statement]] of the definition of [[Definition:Completely Prime Filter|completely prime filter]]:
:$b \notin p$
By the [[Definition:... | Meet Irreducible Element Induced by Completely Prime Filter | https://proofwiki.org/wiki/Meet_Irreducible_Element_Induced_by_Completely_Prime_Filter | https://proofwiki.org/wiki/Meet_Irreducible_Element_Induced_by_Completely_Prime_Filter | [
"Completely Prime Filters",
"Meet Irreducible Elements"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Greatest Element",
"Definition:Completely Prime Filter",
"Definition:Supremum",
"Definition:Meet Irreducible Element",
"Definition:Equals"
] | [
"Definition:Infimum of Set",
"Definition:Contrapositive Statement",
"Definition:Completely Prime Filter",
"Definition:Contrapositive Statement",
"Axiom:Filter Axioms",
"Definition:Supremum of Set",
"Axiom:Ordering Axioms",
"Definition:Meet Irreducible Element",
"Category:Completely Prime Filters",
... |
proofwiki-22328 | Completely Prime Filter Induced by Meet Irreducible Element | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $b$ be a meet-irreducible element of $L$.
Let:
:$p = \set{a \in L : a \npreceq b}$
Then:
:$p$ is a completely prime filter of $L$. | From meet irreducible element, it is sufficient to show:
:$(1)\quad\forall A \subseteq L : \bigvee A \in p \iff \paren{\exists a \in A : a \in p}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigwedge A \in p \iff \paren{\forall a \in A : a \in p}$
where:
:* $\bigvee A$ denotes the supremum of $A$ in $L$
:* $\bigwedge ... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $b$ be a [[Definition:Meet Irreducible Element|meet-irreducible element]] of $L$.
Let:
:$p = \set{a \in L : a \npreceq b}$
Then:
:$p$ is a [[Definition:Completely Prime Filter|completely prime filter]] of $L$. | From [[Definition:Meet Irreducible Element|meet irreducible element]], it is sufficient to show:
:$(1)\quad\forall A \subseteq L : \bigvee A \in p \iff \paren{\exists a \in A : a \in p}$
:$(2)\quad\forall $ [[Definition:Finite|finite]] $A \subseteq L : \bigwedge A \in p \iff \paren{\forall a \in A : a \in p}$
where:
:*... | Completely Prime Filter Induced by Meet Irreducible Element | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Meet_Irreducible_Element | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Meet_Irreducible_Element | [
"Meet Irreducible Elements",
"Completely Prime Filters"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Meet Irreducible Element",
"Definition:Completely Prime Filter"
] | [
"Definition:Meet Irreducible Element",
"Definition:Finite",
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Infimum of Set"
] |
proofwiki-22329 | Characterization of Meet Irreducible Element | Let $\struct{S, \wedge, \preceq}$ be a meet semilattice.
Let $z \in S$.
Then:
:$z$ is meet irreducible
{{iff}}
:$\forall x, y \in S : \leftparen{z \prec x}$ and $\rightparen{z \prec y} \implies \paren{ z \prec x \wedge y}$
where $z \prec x$ denotes that $z \preceq x$ and $z\neq x$. | === Necessary Condition ===
Let $z$ be a meet irreducible element.
Let:
:$x, y \in S : \leftparen{z \prec x}$ and $\rightparen{z \prec y}$
We have:
:$z \ne x$
:$z \ne y$
By definition of meet irreducible element:
:$z \ne x \wedge y$
By definition of infimum:
:$z \preceq x \wedge y$
Hence:
:$z \prec x \wedge y$
The resu... | Let $\struct{S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $z \in S$.
Then:
:$z$ is [[Definition:Meet Irreducible Element|meet irreducible]]
{{iff}}
:$\forall x, y \in S : \leftparen{z \prec x}$ and $\rightparen{z \prec y} \implies \paren{ z \prec x \wedge y}$
where $z \prec x$ deno... | === Necessary Condition ===
Let $z$ be a [[Definition:Meet Irreducible Element|meet irreducible element]].
Let:
:$x, y \in S : \leftparen{z \prec x}$ and $\rightparen{z \prec y}$
We have:
:$z \ne x$
:$z \ne y$
By definition of [[Definition:Meet Irreducible Element|meet irreducible element]]:
:$z \ne x \wedge y$
... | Characterization of Meet Irreducible Element | https://proofwiki.org/wiki/Characterization_of_Meet_Irreducible_Element | https://proofwiki.org/wiki/Characterization_of_Meet_Irreducible_Element | [
"Meet Irreducible Elements"
] | [
"Definition:Meet Semilattice",
"Definition:Meet Irreducible Element"
] | [
"Definition:Meet Irreducible Element",
"Definition:Meet Irreducible Element",
"Definition:Infimum",
"Definition:Infimum",
"Definition:Meet Irreducible Element"
] |
proofwiki-22330 | Characterization of Join Irreducible Element | Let $\struct{S, \vee, \preceq}$ be a join semilattice.
Let $z \in S$.
Then:
:$z$ is join-irreducible
{{iff}}
:$\forall x, y \in S : \leftparen{x \prec z}$ and $\rightparen{y \prec z} \implies \paren{x \vee y \prec z}$
where $x \prec z$ denotes that $x \preceq z$ and $x \neq z$. | By Dual Pairs (Order Theory):
* join semilattice is dual to meet semilattice.
* join irreducible element is dual to meet irreducible element.
* join is dual to meet.
* succeeds is dual to precedes.
Thus the theorem holds by the duality principle applied to Characterization of Meet Irreducible Element.
{{qed}}
Category:... | Let $\struct{S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $z \in S$.
Then:
:$z$ is [[Definition:Join-Irreducible Element|join-irreducible]]
{{iff}}
:$\forall x, y \in S : \leftparen{x \prec z}$ and $\rightparen{y \prec z} \implies \paren{x \vee y \prec z}$
where $x \prec z$ denotes t... | By [[Dual Pairs (Order Theory)]]:
* [[Definition:Join Semilattice|join semilattice]] is dual to [[Definition:Meet Semilattice|meet semilattice]].
* [[Definition:Join Irreducible Element|join irreducible element]] is dual to [[Definition:Meet Irreducible Element|meet irreducible element]].
* [[Definition:Join|join]] is ... | Characterization of Join Irreducible Element | https://proofwiki.org/wiki/Characterization_of_Join_Irreducible_Element | https://proofwiki.org/wiki/Characterization_of_Join_Irreducible_Element | [
"Join Irreducible Elements"
] | [
"Definition:Join Semilattice",
"Definition:Join Irreducible Element"
] | [
"Dual Pairs (Order Theory)",
"Definition:Join Semilattice",
"Definition:Meet Semilattice",
"Definition:Join Irreducible Element",
"Definition:Meet Irreducible Element",
"Definition:Join",
"Definition:Meet",
"Definition:Succeed",
"Definition:Precede",
"Duality Principle (Order Theory)/Global Dualit... |
proofwiki-22331 | Join Prime Element is Join Irreducible | Let $\struct{S, \vee, \preceq}$ be a join semilattice.
Let $z \in S$ be a join-prime element.
Then:
:$z$ is join-irreducible | By Dual Pairs (Order Theory):
* join semilattice is dual to meet semilattice.
* join prime element is dual to meet prime element.
* join irreducible element is dual to meet irreducible element.
Thus the theorem holds by the duality principle applied to Characterization of Meet Irreducible Element.
{{qed}}
Category:Join... | Let $\struct{S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $z \in S$ be a [[Definition:Join Prime Element|join-prime element]].
Then:
:$z$ is [[Definition:Join Irreducible Element|join-irreducible]] | By [[Dual Pairs (Order Theory)]]:
* [[Definition:Join Semilattice|join semilattice]] is dual to [[Definition:Meet Semilattice|meet semilattice]].
* [[Definition:Join Prime Element|join prime element]] is dual to [[Definition:Meet Prime Element|meet prime element]].
* [[Definition:Join Irreducible Element|join irreducib... | Join Prime Element is Join Irreducible | https://proofwiki.org/wiki/Join_Prime_Element_is_Join_Irreducible | https://proofwiki.org/wiki/Join_Prime_Element_is_Join_Irreducible | [
"Join Prime Elements",
"Join Irreducible Elements"
] | [
"Definition:Join Semilattice",
"Definition:Join Prime Element",
"Definition:Join Irreducible Element"
] | [
"Dual Pairs (Order Theory)",
"Definition:Join Semilattice",
"Definition:Meet Semilattice",
"Definition:Join Prime Element",
"Definition:Prime Element (Order Theory)",
"Definition:Join Irreducible Element",
"Definition:Meet Irreducible Element",
"Duality Principle (Order Theory)/Global Duality",
"Cha... |
proofwiki-22332 | Join Prime Element iff Join Irreducible in Distributive Lattice | Let $\struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.
Let $z \in S$.
Then:
:$z$ is join-irreducible
{{iff}}
:$z$ is join-prime | By Dual Pairs (Order Theory):
* join prime element is dual to meet prime element.
* join irreducible element is dual to meet irreducible element.
Thus the theorem holds by the duality principle applied to Characterization of Meet Irreducible Element.
{{qed}}
Category:Join Irreducible Elements
Category:Join Prime Elemen... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Distributive Lattice|distributive lattice]].
Let $z \in S$.
Then:
:$z$ is [[Definition:Join-Irreducible Element|join-irreducible]]
{{iff}}
:$z$ is [[Definition:Join-Prime Element|join-prime]] | By [[Dual Pairs (Order Theory)]]:
* [[Definition:Join Prime Element|join prime element]] is dual to [[Definition:Meet Prime Element|meet prime element]].
* [[Definition:Join Irreducible Element|join irreducible element]] is dual to [[Definition:Meet Irreducible Element|meet irreducible element]].
Thus the theorem hold... | Join Prime Element iff Join Irreducible in Distributive Lattice | https://proofwiki.org/wiki/Join_Prime_Element_iff_Join_Irreducible_in_Distributive_Lattice | https://proofwiki.org/wiki/Join_Prime_Element_iff_Join_Irreducible_in_Distributive_Lattice | [
"Join Irreducible Elements",
"Join Prime Elements",
"Distributive Lattices"
] | [
"Definition:Distributive Lattice",
"Definition:Join Irreducible Element",
"Definition:Join Prime Element"
] | [
"Dual Pairs (Order Theory)",
"Definition:Join Prime Element",
"Definition:Prime Element (Order Theory)",
"Definition:Join Irreducible Element",
"Definition:Meet Irreducible Element",
"Duality Principle (Order Theory)/Global Duality",
"Characterization of Meet Irreducible Element",
"Category:Join Irred... |
proofwiki-22333 | Velocity with respect to Relative Velocity | Let $A$ and $B$ be bodies in space.
Let $\mathbf v_A$ and $\mathbf v_B$ denote the velocities of $A$ and $B$ such that $\mathbf v_A$ and $\mathbf v_B$ are very much smaller than the speed of light.
Let $\mathbf v_{AB}$ denote the velocity of $A$ relative to $B$.
Then:
:$\mathbf v_A = \mathbf v_{AB} + \mathbf v_B$ | {{Recall|Relative Velocity|index = 2}}
{{:Definition:Relative Velocity/Definition 2}}
The result follows directly.
{{qed}} | Let $A$ and $B$ be [[Definition:Body|bodies]] in [[Definition:Ordinary Space|space]].
Let $\mathbf v_A$ and $\mathbf v_B$ denote the [[Definition:Velocity|velocities]] of $A$ and $B$ such that $\mathbf v_A$ and $\mathbf v_B$ are very much smaller than the [[Definition:Speed of Light|speed of light]].
Let $\mathbf v_{... | {{Recall|Relative Velocity|index = 2}}
{{:Definition:Relative Velocity/Definition 2}}
The result follows directly.
{{qed}} | Velocity with respect to Relative Velocity | https://proofwiki.org/wiki/Velocity_with_respect_to_Relative_Velocity | https://proofwiki.org/wiki/Velocity_with_respect_to_Relative_Velocity | [
"Relative Velocity"
] | [
"Definition:Body",
"Definition:Ordinary Space",
"Definition:Velocity",
"Definition:Speed of Light",
"Definition:Relative Velocity"
] | [] |
proofwiki-22334 | Acceleration with respect to Relative Acceleration | Let $A$ and $B$ be bodies in space.
Let $\mathbf a_A$ and $\mathbf a_B$ denote the accelerations of $A$ and $B$.
Let $\mathbf a_{AB}$ denote the acceleration of $A$ relative to $B$.
Then:
:$\mathbf a_A = \mathbf a_{AB} + \mathbf a_B$ | {{Recall|Relative Acceleration}}
{{:Definition:Relative Acceleration}}
The result follows directly.
{{qed}} | Let $A$ and $B$ be [[Definition:Body|bodies]] in [[Definition:Ordinary Space|space]].
Let $\mathbf a_A$ and $\mathbf a_B$ denote the [[Definition:Acceleration|accelerations]] of $A$ and $B$.
Let $\mathbf a_{AB}$ denote the [[Definition:Relative Acceleration|acceleration of $A$ relative to $B$]].
Then:
:$\mathbf a_A ... | {{Recall|Relative Acceleration}}
{{:Definition:Relative Acceleration}}
The result follows directly.
{{qed}} | Acceleration with respect to Relative Acceleration | https://proofwiki.org/wiki/Acceleration_with_respect_to_Relative_Acceleration | https://proofwiki.org/wiki/Acceleration_with_respect_to_Relative_Acceleration | [
"Relative Velocity"
] | [
"Definition:Body",
"Definition:Ordinary Space",
"Definition:Acceleration",
"Definition:Relative Acceleration"
] | [] |
proofwiki-22335 | Characterization of Meet-Irreducible Open Set | Let $\struct {S, \tau}$ be a topological space.
Let $W \in \tau$.
{{TFAE}}:
:$(1)\quad W$ is a meet-irreducible open set
:$(2)\quad \forall U, V \in \tau : \paren {U \cap V = W \implies U = W \text { or } V = W}$
:$(3)\quad \forall U, V \in \tau : \paren {U \cap V \subseteq W \implies U \subseteq W \text { or } V \subs... | By definition of meet-irreducible open set:
:$W$ is meet-irreducible open set
{{iff}}:
:$W$ is meet-irreducible in the frame $\struct {\tau, \subseteq}$
By definition of meet-irreducible:
:$W$ is meet-irreducible in the frame $\struct {\tau, \subseteq}$
{{iff}}:
:$\forall U, V \in \tau : \paren {U \cap V = W \implies... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $W \in \tau$.
{{TFAE}}:
:$(1)\quad W$ is a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]
:$(2)\quad \forall U, V \in \tau : \paren {U \cap V = W \implies U = W \text { or } V = W}$
:$(3)\quad \forall U, V \in \t... | By definition of [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]:
:$W$ is [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]
{{iff}}:
:$W$ is [[Definition:Meet-Irreducible Element|meet-irreducible]] in the [[Definition:Frame (Lattice Theory)|frame]] $\struct {\tau, \subseteq}$
By d... | Characterization of Meet-Irreducible Open Set | https://proofwiki.org/wiki/Characterization_of_Meet-Irreducible_Open_Set | https://proofwiki.org/wiki/Characterization_of_Meet-Irreducible_Open_Set | [
"Meet-Irreducible Open Sets"
] | [
"Definition:Topological Space",
"Definition:Meet-Irreducible Open Set"
] | [
"Definition:Meet-Irreducible Open Set",
"Definition:Meet-Irreducible Open Set",
"Definition:Meet Irreducible Element",
"Definition:Frame (Lattice Theory)",
"Definition:Meet Irreducible Element",
"Definition:Meet Irreducible Element",
"Definition:Frame (Lattice Theory)",
"Prime Element iff Meet Irreduc... |
proofwiki-22336 | Babczyński Theorem | Let $n$ be a $6$-digit integer of the form $\sqbrk {xyxyxy}$ for digits $x$ and $y$.
Then $n$ is divisible by $3$, $7$, $13$ and $37$. | {{begin-eqn}}
{{eqn | l = \sqbrk {xyxyxy}
| r = \sqbrk {xy} \times 10101
| c =
}}
{{eqn | r = \sqbrk {xy} \times 3 \times 7 \times 13 \times 37
| c =
}}
{{end-eqn}}
{{qed}} | Let $n$ be a $6$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] of the form $\sqbrk {xyxyxy}$ for [[Definition:Digit|digits]] $x$ and $y$.
Then $n$ is [[Definition:Divisor of Integer|divisible]] by $3$, $7$, $13$ and $37$. | {{begin-eqn}}
{{eqn | l = \sqbrk {xyxyxy}
| r = \sqbrk {xy} \times 10101
| c =
}}
{{eqn | r = \sqbrk {xy} \times 3 \times 7 \times 13 \times 37
| c =
}}
{{end-eqn}}
{{qed}} | Babczyński Theorem | https://proofwiki.org/wiki/Babczyński_Theorem | https://proofwiki.org/wiki/Babczyński_Theorem | [
"Babczyński Theorem",
"3",
"7",
"13",
"37",
"Divisibility"
] | [
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-22337 | Atom of Countably Generated Sigma-Algebra is Measurable | Let $X$ be a set.
Let $\Sigma$ be a countably generated $\sigma$-algebra of $X$.
Then:
:$\ds \forall x \in X : \quad \sqbrk x_\Sigma \in \Sigma$
where $\sqbrk x_\Sigma$ denotes the atom. | By assumption, there is a countable collection $\GG \subseteq \Sigma$ such that:
:$\ds \Sigma = \map \sigma \GG$
Let $x \in X$.
It suffices to show:
:$\ds \sqbrk x_\Sigma = \bigcap_{ x \in A \in \GG } A \cap \bigcap_{ x \not \in A \in \GG} X \setminus A$
$\subseteq$ is clear.
We need to show $\supseteq$.
{{ProofWanted}... | Let $X$ be a [[Definition:Set|set]].
Let $\Sigma$ be a [[Definition:Countably Generated Sigma-Algebra|countably generated $\sigma$-algebra]] of $X$.
Then:
:$\ds \forall x \in X : \quad \sqbrk x_\Sigma \in \Sigma$
where $\sqbrk x_\Sigma$ denotes the [[Definition:Atom of Countably Generated Sigma-Algebra|atom]]. | By assumption, there is a [[Definition:Countable Set|countable collection]] $\GG \subseteq \Sigma$ such that:
:$\ds \Sigma = \map \sigma \GG$
Let $x \in X$.
It suffices to show:
:$\ds \sqbrk x_\Sigma = \bigcap_{ x \in A \in \GG } A \cap \bigcap_{ x \not \in A \in \GG} X \setminus A$
$\subseteq$ is clear.
We need to... | Atom of Countably Generated Sigma-Algebra is Measurable | https://proofwiki.org/wiki/Atom_of_Countably_Generated_Sigma-Algebra_is_Measurable | https://proofwiki.org/wiki/Atom_of_Countably_Generated_Sigma-Algebra_is_Measurable | [
"Definitions/Sigma-Algebras"
] | [
"Definition:Set",
"Definition:Countably Generated Sigma-Algebra",
"Definition:Atom of Countably Generated Sigma-Algebra"
] | [
"Definition:Countable Set",
"Category:Definitions/Sigma-Algebras"
] |
proofwiki-22338 | Join Irreducible Element is Dual of Meet Irreducible Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $z \in S$.
The following are dual statements:
:$z$ is a meet irreducible element of the meet semilattice $\struct {S, \wedge, \preceq}$
:$z$ is a join irreducible element of the join semilattice $\struct {S, \vee, \preceq}$ | By definition of meet irreducible element:
:$z$ is the meet irreducible element of the meet semilattice $\struct {S, \wedge, \preceq}$
{{iff}}:
:$\forall x, y \in S: \paren{z = x \wedge y} \implies \leftparen{z = x}$ or $\rightparen{z = y}$
By the duality principle, the dual of this statement is:
:$\forall x, y \in S: ... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $z \in S$.
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$z$ is a [[Definition:Meet Irreducible Element|meet irreducible element]] of the [[Definition:Meet Semilattice|meet semilattice]] $\struct {S, \wed... | By definition of [[Definition:Meet Irreducible Element|meet irreducible element]]:
:$z$ is the [[Definition:Meet Irreducible Element|meet irreducible element]] of the [[Definition:Meet Semilattice|meet semilattice]] $\struct {S, \wedge, \preceq}$
{{iff}}:
:$\forall x, y \in S: \paren{z = x \wedge y} \implies \leftparen... | Join Irreducible Element is Dual of Meet Irreducible Element | https://proofwiki.org/wiki/Join_Irreducible_Element_is_Dual_of_Meet_Irreducible_Element | https://proofwiki.org/wiki/Join_Irreducible_Element_is_Dual_of_Meet_Irreducible_Element | [
"Meet Irreducible Elements",
"Join Irreducible Elements",
"Dual Pairs (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Dual Statement (Order Theory)",
"Definition:Meet Irreducible Element",
"Definition:Meet Semilattice",
"Definition:Join Irreducible Element",
"Definition:Join Semilattice"
] | [
"Definition:Meet Irreducible Element",
"Definition:Meet Irreducible Element",
"Definition:Meet Semilattice",
"Duality Principle (Order Theory)",
"Definition:Dual Statement (Order Theory)",
"Definition:Join Irreducible Element",
"Definition:Join Irreducible Element",
"Definition:Join Semilattice"
] |
proofwiki-22339 | Join Prime Element is Dual of Meet Prime Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $z \in S$.
The following are dual statements:
:$z$ is the meet prime element of the meet semilattice $\struct {S, \wedge, \preceq}$
:$z$ is the join prime element of the join semilattice $\struct {S, \vee, \preceq}$ | By definition of meet prime element:
:$z$ is the meet prime element of the meet semilattice $\struct {S, \wedge, \preceq}$
{{iff}}:
:$\forall x, y \in S: \paren{x \wedge y \preceq z} \implies \leftparen{x \preceq z}$ or $\rightparen{y \preceq z}$
By the duality principle, the dual of this statement is:
:$\forall x, y \... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $z \in S$.
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$z$ is the [[Definition:Meet Prime Element|meet prime element]] of the [[Definition:Meet Semilattice|meet semilattice]] $\struct {S, \wedge, \prece... | By definition of [[Definition:Meet Prime Element|meet prime element]]:
:$z$ is the [[Definition:Meet Prime Element|meet prime element]] of the [[Definition:Meet Semilattice|meet semilattice]] $\struct {S, \wedge, \preceq}$
{{iff}}:
:$\forall x, y \in S: \paren{x \wedge y \preceq z} \implies \leftparen{x \preceq z}$ or ... | Join Prime Element is Dual of Meet Prime Element | https://proofwiki.org/wiki/Join_Prime_Element_is_Dual_of_Meet_Prime_Element | https://proofwiki.org/wiki/Join_Prime_Element_is_Dual_of_Meet_Prime_Element | [
"Prime Elements",
"Join Prime Elements",
"Dual Pairs (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Dual Statement (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Meet Semilattice",
"Definition:Join Prime Element",
"Definition:Join Semilattice"
] | [
"Definition:Prime Element (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Meet Semilattice",
"Duality Principle (Order Theory)",
"Definition:Dual Statement (Order Theory)",
"Definition:Join Prime Element",
"Definition:Join Prime Element",
"Definition:Join Semilattice"
] |
proofwiki-22340 | Generator of Sigma-Algebra Separates Points | Let $X$ be a set.
Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.
Let $\map \sigma \GG$ be the $\sigma$-algebra generated by $\GG$.
Let $x, y \in X$.
Suppose that $\map \sigma \GG$ separates $x$ and $y$ in the sense:
:$\exists A \in \map \sigma \GG : \quad x \in A \quad \wedge \quad y \in X \setminu... | {{AimForCont}}:
:$\forall A_0 \in \GG : \quad x, y \in A_0 \quad \vee \quad x, y \in X \setminus A_0$
Let:
:$\CC := \set { A \in \map \sigma {\GG} : \quad x, y \in A \quad \vee \quad x, y \in X \setminus A }$
Then $\CC$ is a $\sigma$-algebra.
Since:
:$\GG \subseteq \CC$
we have:
:$\map \sigma \GG \subseteq \CC$
This ... | Let $X$ be a [[Definition:Set|set]].
Let $\GG \subseteq \powerset X$ be a [[Definition:Set|collection]] of [[Definition:Subset|subsets]] of $X$.
Let $\map \sigma \GG$ be the [[Definition:Sigma-Algebra Generated by Collection of Subsets|$\sigma$-algebra generated by $\GG$]].
Let $x, y \in X$.
Suppose that $\map \sig... | {{AimForCont}}:
:$\forall A_0 \in \GG : \quad x, y \in A_0 \quad \vee \quad x, y \in X \setminus A_0$
Let:
:$\CC := \set { A \in \map \sigma {\GG} : \quad x, y \in A \quad \vee \quad x, y \in X \setminus A }$
Then $\CC$ is a [[Definition:Sigma-Algebra|$\sigma$-algebra]].
Since:
:$\GG \subseteq \CC$
we have:
:$\map... | Generator of Sigma-Algebra Separates Points | https://proofwiki.org/wiki/Generator_of_Sigma-Algebra_Separates_Points | https://proofwiki.org/wiki/Generator_of_Sigma-Algebra_Separates_Points | [
"Definitions/Sigma-Algebras"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Subset",
"Definition:Sigma-Algebra Generated by Collection of Subsets"
] | [
"Definition:Sigma-Algebra",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Definitions/Sigma-Algebras"
] |
proofwiki-22341 | Characterization of Supremum Precedes Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ admit a supremum $a$
Let $b \in S$.
Then:
:$a \preceq b$
{{iff}}:
:$\forall t \in T : t \preceq b$ | === Necessary Condition ===
Let $a \preceq b$.
By definition of supremum:
:$a$ is an upper bound for $T$
By definition of upper bound:
:$\forall t \in T : t \preceq a$
From {{Ordering-axiom|2}}:
:$\forall t \in T : t \preceq b$
{{qed|lemma}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T \subseteq S$ admit a [[Definition:Supremum of Set|supremum]] $a$
Let $b \in S$.
Then:
:$a \preceq b$
{{iff}}:
:$\forall t \in T : t \preceq b$ | === Necessary Condition ===
Let $a \preceq b$.
By definition of [[Definition:Supremum of Set|supremum]]:
:$a$ is an [[Definition:Upper Bound|upper bound]] for $T$
By definition of [[Definition:Upper Bound|upper bound]]:
:$\forall t \in T : t \preceq a$
From {{Ordering-axiom|2}}:
:$\forall t \in T : t \preceq b$
{{q... | Characterization of Supremum Precedes Element | https://proofwiki.org/wiki/Characterization_of_Supremum_Precedes_Element | https://proofwiki.org/wiki/Characterization_of_Supremum_Precedes_Element | [
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set"
] | [
"Definition:Supremum of Set",
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Supremum of Set"
] |
proofwiki-22342 | Integral of Integrable Function is Additive/Complex Function | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $f, g : X \to \C$ be a $\mu$-integrable function.
Then $f + g$ is $\mu$-integrable and:
:$\ds \int \paren {f + g} \rd \mu = \int f \rd \mu + \int g ... | From Addition of Real and Imaginary Parts, we have:
:$\map \Re {f + g} = \map \Re f + \map \Re g$
and:
:$\map \Im {f + g} = \map \Im f + \map \Im g$
Since $f$ is $\mu$-integrable:
:$\map \Re f$ and $\map \Im f$ are $\mu$-integrable.
Similarly since $g$ is $\mu$-integrable:
:$\map \Re g$ and $\map \Im g$ are $\mu$-integ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\struct {\C, \map \BB \C}$ be the [[Definition:Complex Number|complex numbers]] made into a [[Definition:Measurable Space|measurable space]] with its [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]].
Let $f, g : X \to \C$ be... | From [[Addition of Real and Imaginary Parts]], we have:
:$\map \Re {f + g} = \map \Re f + \map \Re g$
and:
:$\map \Im {f + g} = \map \Im f + \map \Im g$
Since $f$ is [[Definition:Complex Measure-Integrable Function|$\mu$-integrable]]:
:$\map \Re f$ and $\map \Im f$ are [[Definition:Measure-Integrable Function|$\mu$-in... | Integral of Integrable Function is Additive/Complex Function | https://proofwiki.org/wiki/Integral_of_Integrable_Function_is_Additive/Complex_Function | https://proofwiki.org/wiki/Integral_of_Integrable_Function_is_Additive/Complex_Function | [
"Integral of Integrable Function is Additive",
"Complex Measure-Integrable Functions"
] | [
"Definition:Measure Space",
"Definition:Complex Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function/Measure Space/Complex Function"
] | [
"Addition of Real and Imaginary Parts",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function/Measure Space",
"Integral of Integrable Function is Addi... |
proofwiki-22343 | Integral of Integrable Function is Homogeneous/Complex Function | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $f : X \to \C$ be a $\mu$-integrable function.
Let $\lambda \in \C$.
Then $\lambda f$ is $\mu$-integrable and:
:$\ds \int \lambda f \rd \mu = \lamb... | We have:
{{begin-eqn}}
{{eqn | l = \lambda \int f \rd \mu
| r = \paren {\map \Re \lambda + i \Im \lambda} \int \map \Re f \rd \mu + i \paren {\map \Re \lambda + i \map \Im \lambda} \int \Im f \rd \mu
}}
{{eqn | r = \paren {\map \Re \lambda \int \map \Re f \rd \mu - \map \Im \lambda \int \map \Im f \rd \mu} + i \pare... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\struct {\C, \map \BB \C}$ be the [[Definition:Complex Number|complex numbers]] made into a [[Definition:Measurable Space|measurable space]] with its [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]].
Let $f : X \to \C$ be a ... | We have:
{{begin-eqn}}
{{eqn | l = \lambda \int f \rd \mu
| r = \paren {\map \Re \lambda + i \Im \lambda} \int \map \Re f \rd \mu + i \paren {\map \Re \lambda + i \map \Im \lambda} \int \Im f \rd \mu
}}
{{eqn | r = \paren {\map \Re \lambda \int \map \Re f \rd \mu - \map \Im \lambda \int \map \Im f \rd \mu} + i \pare... | Integral of Integrable Function is Homogeneous/Complex Function | https://proofwiki.org/wiki/Integral_of_Integrable_Function_is_Homogeneous/Complex_Function | https://proofwiki.org/wiki/Integral_of_Integrable_Function_is_Homogeneous/Complex_Function | [
"Integral of Integrable Function is Homogeneous"
] | [
"Definition:Measure Space",
"Definition:Complex Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function/Measure Space/Complex Function"
] | [
"Real Part of Complex Product",
"Imaginary Part of Complex Product",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function/Measure Space",
"Integral of Integrable Function is Additive",
"Integral of Integrable Function is Homogeneous",
"Definition:Integrable Fu... |
proofwiki-22344 | Complex Modulus of Measurable Function is Measurable | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $f : X \to \C$ be a $\Sigma/\map \BB \C$-measurable mapping.
Let $\struct {\R, \map \BB \R}$ be the real numbers made into a measurable space with i... | From Complex Modulus Function is Continuous, the mapping $\cmod {\, \cdot \,} : \C \to \R$ is continuous.
From Continuous Mapping is Measurable, $\cmod {\, \cdot \,} : \C \to \R$ is $\map \BB C/\map \BB \R$-measurable.
Hence from Composition of Measurable Mappings is Measurable, $\cmod f : X \to \R$ is $\Sigma/\map \BB... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\struct {\C, \map \BB \C}$ be the [[Definition:Complex Number|complex numbers]] made into a [[Definition:Measurable Space|measurable space]] with its [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]].
Let $f : X \to \C$ be a ... | From [[Complex Modulus Function is Continuous]], the [[Definition:Mapping|mapping]] $\cmod {\, \cdot \,} : \C \to \R$ is [[Definition:Continuous Function|continuous]].
From [[Continuous Mapping is Measurable]], $\cmod {\, \cdot \,} : \C \to \R$ is [[Definition:Measurable Mapping|$\map \BB C/\map \BB \R$-measurable]].
... | Complex Modulus of Measurable Function is Measurable | https://proofwiki.org/wiki/Complex_Modulus_of_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Complex_Modulus_of_Measurable_Function_is_Measurable | [
"Measurable Mappings",
"Measurable Functions",
"Measurable Functions"
] | [
"Definition:Measure Space",
"Definition:Complex Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Measurable Mapping",
"Definition:Real Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Definition:Measurable Mapping"
] | [
"Complex Modulus Function is Continuous",
"Definition:Mapping",
"Definition:Continuous Function",
"Continuous Mapping is Measurable",
"Definition:Measurable Mapping",
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Mapping",
"Category:Measurable Functions"
] |
proofwiki-22345 | Existence of Smooth Transition Function in One Dimension/Lemma | $f$ is smooth with:
:$\ds \map {f^{(n)} } x = \begin{cases}\map {P_n} {\frac 1 x} e^{-1/x} & x > 0 \\ 0 & x \le 0\end{cases}$
for each $n \ge 1$, for some polynomial $P_n$. | We proceed by Proof by Mathematical Induction.
For $n \ge 1$, let $\map Q n$ be the proposition:
:$f$ is $n$ times differentiable and there exists a polynomial $P_n$ with positive leading coefficient such that:
::$\ds \map {f^{(n)} } x = \begin{cases}\map {P_n} {\frac 1 x} e^{-1/x} & x > 0 \\ 0 & x \le 0\end{cases}$ | $f$ is [[Definition:Smooth Function|smooth]] with:
:$\ds \map {f^{(n)} } x = \begin{cases}\map {P_n} {\frac 1 x} e^{-1/x} & x > 0 \\ 0 & x \le 0\end{cases}$
for each $n \ge 1$, for some [[Definition:Polynomial|polynomial]] $P_n$. | We proceed by [[Proof by Mathematical Induction]].
For $n \ge 1$, let $\map Q n$ be the proposition:
:$f$ is $n$ times [[Definition:Differentiable Function|differentiable]] and there exists a [[Definition:Polynomial|polynomial]] $P_n$ with positive leading coefficient such that:
::$\ds \map {f^{(n)} } x = \begin{cases... | Existence of Smooth Transition Function in One Dimension/Lemma | https://proofwiki.org/wiki/Existence_of_Smooth_Transition_Function_in_One_Dimension/Lemma | https://proofwiki.org/wiki/Existence_of_Smooth_Transition_Function_in_One_Dimension/Lemma | [
"Existence of Smooth Transition Function in One Dimension"
] | [
"Definition:Smooth Real Function",
"Definition:Polynomial"
] | [
"Principle of Mathematical Induction",
"Definition:Differentiable Mapping",
"Definition:Polynomial",
"Definition:Differentiable Mapping",
"Definition:Differentiable Mapping",
"Definition:Polynomial",
"Definition:Differentiable Mapping"
] |
proofwiki-22346 | Existence of Smooth Transition Function in One Dimension | Let $a, b, c, d \in \R$ be real numbers with $a < b < c < d$.
Then there exists a smooth function $h : \R \to \closedint 0 1$ such that:
:$\map h x = 1$ for $x \in \closedint b c$
and:
:$\map h x = 0$ for $x \in \R \setminus \openint a d$ | First define $f : \R \to \R$ by:
:$\ds \map f x = \begin{cases}e^{-1/x} & x \ge 0 \\ 0 & x < 0\end{cases}$
for each $x \in \R$.
Note that $\map f x \ge 0$ for all $x \in \R$.
We first show that $f$ is smooth. | Let $a, b, c, d \in \R$ be [[Definition:Real Number|real numbers]] with $a < b < c < d$.
Then there exists a [[Definition:Smooth Function|smooth function]] $h : \R \to \closedint 0 1$ such that:
:$\map h x = 1$ for $x \in \closedint b c$
and:
:$\map h x = 0$ for $x \in \R \setminus \openint a d$ | First define $f : \R \to \R$ by:
:$\ds \map f x = \begin{cases}e^{-1/x} & x \ge 0 \\ 0 & x < 0\end{cases}$
for each $x \in \R$.
Note that $\map f x \ge 0$ for all $x \in \R$.
We first show that $f$ is [[Definition:Smooth Function|smooth]]. | Existence of Smooth Transition Function in One Dimension | https://proofwiki.org/wiki/Existence_of_Smooth_Transition_Function_in_One_Dimension | https://proofwiki.org/wiki/Existence_of_Smooth_Transition_Function_in_One_Dimension | [
"Existence of Smooth Transition Function in One Dimension"
] | [
"Definition:Real Number",
"Definition:Smooth Real Function"
] | [
"Definition:Smooth Real Function",
"Definition:Smooth Real Function",
"Definition:Smooth Real Function"
] |
proofwiki-22347 | Limit of Countable Union of Unbounded Below Closed Intervals | Let $x \in \R$.
,
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence converging to $x$ such that $x_n < x$ for each $n \in \N$.
Then we have:
:$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \openint {-\infty} x$ | First suppose that:
:$\ds t \in \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then we have:
:$t \le x_n$ for some $n \in \N$.
We have $x_n < x$ for each $n \in \N$.
Hence $t < x$ and $t \in \openint {-\infty} x$.
Now suppose that $t \in \openint {-\infty} x$.
Then $t < x$.
Hence we can take $\epsilon > 0$ su... | Let $x \in \R$.
,
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] [[Definition:Convergent Real Sequence|converging]] to $x$ such that $x_n < x$ for each $n \in \N$.
Then we have:
:$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \openint {-\infty} x$ | First suppose that:
:$\ds t \in \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then we have:
:$t \le x_n$ for some $n \in \N$.
We have $x_n < x$ for each $n \in \N$.
Hence $t < x$ and $t \in \openint {-\infty} x$.
Now suppose that $t \in \openint {-\infty} x$.
Then $t < x$.
Hence we can take $\epsilon ... | Limit of Countable Union of Unbounded Below Closed Intervals | https://proofwiki.org/wiki/Limit_of_Countable_Union_of_Unbounded_Below_Closed_Intervals | https://proofwiki.org/wiki/Limit_of_Countable_Union_of_Unbounded_Below_Closed_Intervals | [
"Limits of Sequences of Intervals"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Category:Limits of Sequences of Intervals"
] |
proofwiki-22348 | Monotonic Sequence Characterization of Left Limit of Function | Let $\hointl a b$ be a real interval.
Let $x \in \hointl a b$.
Let $f : \hointl a b \to \R$ be a real function.
Let $L \in \R$.
Then:
:$\ds \lim_{y \mathop \to x^-} \map f y = L$
{{iff}}:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n < x$ for each $n$, that converge to $x$ we have:
::$... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{y \to x^-} \map f y = L$
then:
:for each real sequence $\sequence {x_n}_{n \mathop \in \N}$, with $x_n < x$ for each $n$, converging to $x$ we have:
::$\map f {x_n} \to L$
from Limit of Function by Convergent Sequences: Corollary.
So in particular:
:for all monoton... | Let $\hointl a b$ be a [[Definition:Real Interval|real interval]].
Let $x \in \hointl a b$.
Let $f : \hointl a b \to \R$ be a [[Definition:Real Function|real function]].
Let $L \in \R$.
Then:
:$\ds \lim_{y \mathop \to x^-} \map f y = L$
{{iff}}:
:for all [[Definition:Monotone Real Sequence|monotone sequences]]... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{y \to x^-} \map f y = L$
then:
:for each [[Definition:Real Sequence|real sequence]] $\sequence {x_n}_{n \mathop \in \N}$, with $x_n < x$ for each $n$, [[Definition:Convergent Real Sequence|converging]] to $x$ we have:
::$\map f {x_n} \to L$
from [[Limit of Function... | Monotonic Sequence Characterization of Left Limit of Function | https://proofwiki.org/wiki/Monotonic_Sequence_Characterization_of_Left_Limit_of_Function | https://proofwiki.org/wiki/Monotonic_Sequence_Characterization_of_Left_Limit_of_Function | [
"Monotonic Sequence Characterization of Left Limit of Function"
] | [
"Definition:Real Interval",
"Definition:Real Function",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Limit of Function by Convergent Sequences/Corollary",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Defi... |
proofwiki-22349 | Left Limit of Distribution Function of Finite Borel Measure | Let $\mu$ be a finite Borel measure.
Let $F_\mu$ be the distribution function of $\mu$.
Then we have:
:$\ds \lim_{y \mathop \to x^-} \map {F_\mu} y = \map \mu {\openint {-\infty} x}$ | From Monotonic Sequence Characterization of Left Limit of Function, the claim is equivalent to:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n < x$ for each $n$, that converge to $x$ we have:
::$\map {F_\mu} {x_n} \to \map \mu {\openint {-\infty} x}$
That is:
:$\map \mu {\hointl {-\infty} {... | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]].
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Then we have:
:$\ds \lim_{y \mathop \to x^-} \map {F_\mu} y = \map \mu {\openint {-\infty} x}$ | From [[Monotonic Sequence Characterization of Left Limit of Function]], the claim is equivalent to:
:for all [[Definition:Monotone Real Sequence|monotone sequences]] $\sequence {x_n}_{n \mathop \in \N}$, with $x_n < x$ for each $n$, that [[Definition:Convergent Real Sequence|converge]] to $x$ we have:
::$\map {F_\mu} {... | Left Limit of Distribution Function of Finite Borel Measure | https://proofwiki.org/wiki/Left_Limit_of_Distribution_Function_of_Finite_Borel_Measure | https://proofwiki.org/wiki/Left_Limit_of_Distribution_Function_of_Finite_Borel_Measure | [
"Distribution Functions of Finite Borel Measures",
"Distribution Function of Finite Borel Measure",
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure"
] | [
"Monotonic Sequence Characterization of Left Limit of Function",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Limit of Countable Union of Unbounded Below Closed Intervals",
"Definit... |
proofwiki-22350 | Distribution Function of Finite Borel Measure is Continuous at Point iff Measure Continuous at Point | Let $\mu$ be a finite Borel measure.
Let $F_\mu$ be the distribution function of $\mu$.
Let $x \in \R$.
Then $F_\mu$ is continuous at $x$ {{iff}} $\map \mu {\set x} = 0$. | From Distribution Function of Finite Borel Measure is Right-Continuous, $F_\mu$ is right-continuous at $x$.
Hence from Continuous at Point iff Left-Continuous and Right-Continuous, the claim is equivalent to:
:$F_\mu$ is left-continuous at $x$ {{iff}} $\map \mu {\set x} = 0$.
That is:
:$\ds \lim_{y \to x^-} \map {F_\mu... | Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]].
Let $F_\mu$ be the [[Definition:Distribution Function of Finite Borel Measure|distribution function]] of $\mu$.
Let $x \in \R$.
Then $F_\mu$ is [[Definition:Continuous Function|continuous]] at $x$ {{iff}} $\map \mu {\set... | From [[Distribution Function of Finite Borel Measure is Right-Continuous]], $F_\mu$ is [[Definition:Right-Continuous Real Function|right-continuous]] at $x$.
Hence from [[Continuous at Point iff Left-Continuous and Right-Continuous]], the claim is equivalent to:
:$F_\mu$ is [[Definition:Left-Continuous Real Function|l... | Distribution Function of Finite Borel Measure is Continuous at Point iff Measure Continuous at Point | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Continuous_at_Point_iff_Measure_Continuous_at_Point | https://proofwiki.org/wiki/Distribution_Function_of_Finite_Borel_Measure_is_Continuous_at_Point_iff_Measure_Continuous_at_Point | [
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure",
"Definition:Continuous Function"
] | [
"Distribution Function of Finite Borel Measure is Right-Continuous",
"Definition:Continuous Real Function/Right-Continuous",
"Continuous at Point iff Left-Continuous and Right-Continuous",
"Definition:Continuous Real Function/Left-Continuous",
"Left Limit of Distribution Function of Finite Borel Measure",
... |
proofwiki-22351 | Holomorphic Function with Constant Imaginary Part is Constant | Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.
Let $f: D \to \C$ be a holomorphic function such that $\map \Im f$ is constant.
Then $f$ is constant. | Define $u : \R^2 \to \R$ and $v : \R^2 \to \R$ such that:
:$\map f {x + i y} = \map u {x, y} + i \map v {x, y}$
for each $x, y \in \R$.
By hypothesis, $v$ is constant.
Hence from Derivative of Constant, we have:
:$\ds \frac {\partial v} {\partial x} = 0$
and:
:$\ds \frac {\partial v} {\partial y} = 0$
Hence by the Cau... | Let $D \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$.
Let $f: D \to \C$ be a [[Definition:Holomorphic Function|holomorphic function]] such that $\map \Im f$ is [[Definition:Constant Ma... | Define $u : \R^2 \to \R$ and $v : \R^2 \to \R$ such that:
:$\map f {x + i y} = \map u {x, y} + i \map v {x, y}$
for each $x, y \in \R$.
By hypothesis, $v$ is [[Definition:Constant Mapping|constant]].
Hence from [[Derivative of Constant]], we have:
:$\ds \frac {\partial v} {\partial x} = 0$
and:
:$\ds \frac {\partial... | Holomorphic Function with Constant Imaginary Part is Constant | https://proofwiki.org/wiki/Holomorphic_Function_with_Constant_Imaginary_Part_is_Constant | https://proofwiki.org/wiki/Holomorphic_Function_with_Constant_Imaginary_Part_is_Constant | [
"Holomorphic Function with Constant Imaginary Part is Constant",
"Holomorphic Functions",
"Holomorphic Function with Constant Imaginary Part is Constant"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Holomorphic Function",
"Definition:Constant Mapping",
"Definition:Constant Mapping"
] | [
"Definition:Constant Mapping",
"Derivative of Constant",
"Cauchy-Riemann Equations",
"Zero Derivative implies Constant Function",
"Zero Derivative implies Constant Function",
"Definition:Constant Mapping",
"Definition:Constant Mapping",
"Definition:Constant Mapping",
"Category:Holomorphic Functions"... |
proofwiki-22352 | Holomorphic Function with Constant Imaginary Part is Constant/Corollary | Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.
Let $f: D \to \C$ be a holomorphic function such that $\map \Re f$ is constant.
Then $f$ is constant. | From Combination Theorem for Complex Derivatives: Multiple Rule, $i f$ is holomorphic.
From Imaginary Part of Imaginary Unit times Element of *-Algebra, we have:
:$\map \Im {i f} = \map \Re f$
Hence $\map \Im {i f}$ is constant.
From Holomorphic Function with Constant Imaginary Part is Constant, there exists $c \in \C... | Let $D \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$.
Let $f: D \to \C$ be a [[Definition:Holomorphic Function|holomorphic function]] such that $\map \Re f$ is [[Definition:Constant Ma... | From [[Combination Theorem for Complex Derivatives/Multiple Rule|Combination Theorem for Complex Derivatives: Multiple Rule]], $i f$ is [[Definition:Holomorphic Function|holomorphic]].
From [[Imaginary Part of Imaginary Unit times Element of *-Algebra]], we have:
:$\map \Im {i f} = \map \Re f$
Hence $\map \Im {i f}$... | Holomorphic Function with Constant Imaginary Part is Constant/Corollary | https://proofwiki.org/wiki/Holomorphic_Function_with_Constant_Imaginary_Part_is_Constant/Corollary | https://proofwiki.org/wiki/Holomorphic_Function_with_Constant_Imaginary_Part_is_Constant/Corollary | [
"Holomorphic Function with Constant Imaginary Part is Constant"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Holomorphic Function",
"Definition:Constant Mapping",
"Definition:Constant Mapping"
] | [
"Combination Theorem for Complex Derivatives/Multiple Rule",
"Definition:Holomorphic Function",
"Imaginary Part of Imaginary Unit times Element of *-Algebra",
"Definition:Constant Mapping",
"Holomorphic Function with Constant Imaginary Part is Constant",
"Definition:Constant Mapping",
"Category:Holomorp... |
proofwiki-22353 | Holomorphic Function is Identified by Real or Imaginary Part | Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.
Let $f, g : D \to \C$ be holomorphic functions such that either:
:$\map \Re f = \map \Re g$
or:
:$\map \Im f = \map \Im g$
Then $f = g$. | From Combination Theorem for Complex Derivatives: Combined Sum Rule, we have:
:$f - g$ is holomorphic.
Suppose first that:
:$\map \Re f = \map \Re g$
Then:
:$\map \Re {f - g} = 0$
Hence from {{Corollary|Holomorphic Function with Constant Imaginary Part is Constant}}, we have $f = g$.
Suppose also that:
:$\map \Im f = ... | Let $D \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$.
Let $f, g : D \to \C$ be [[Definition:Holomorphic Function|holomorphic functions]] such that either:
:$\map \Re f = \map \Re g$
or... | From [[Combination Theorem for Complex Derivatives/Combined Sum Rule|Combination Theorem for Complex Derivatives: Combined Sum Rule]], we have:
:$f - g$ is [[Definition:Holomorphic Function|holomorphic]].
Suppose first that:
:$\map \Re f = \map \Re g$
Then:
:$\map \Re {f - g} = 0$
Hence from {{Corollary|Holomorphic ... | Holomorphic Function is Identified by Real or Imaginary Part | https://proofwiki.org/wiki/Holomorphic_Function_is_Identified_by_Real_or_Imaginary_Part | https://proofwiki.org/wiki/Holomorphic_Function_is_Identified_by_Real_or_Imaginary_Part | [
"Holomorphic Functions"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Holomorphic Function"
] | [
"Combination Theorem for Complex Derivatives/Combined Sum Rule",
"Definition:Holomorphic Function",
"Holomorphic Function with Constant Imaginary Part is Constant",
"Category:Holomorphic Functions"
] |
proofwiki-22354 | Completely Prime Filter Induced by Frame Homomorphism Induced by Completely Prime Filter | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the (Boolean Lattice) $\mathbf 2$.
For every completely prime filter $p$ of $L$, let:
:$\phi_p : L \to \mathbf 2$ be the frame homomorphism defined by:
::<nowiki>$\forall a \in L : \map {\phi_p} a = \begin{cases}
... | From Frame Homomorphism Onto Two Induced by Completely Prime Filter:
:For every completely prime filter $p$ of $L$, $\phi_p : L \to \mathbf 2$ is a frame homomorphism
From Completely Prime Filter Induced by Frame Homomorphism Onto Two:
:For every frame homomorphism $\phi : L \to \mathbf 2$, $p_\phi = \map {\phi^{-1}} \... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]].
For every [[Definition:Completely Prime Filter|completely prime filter]] $p$ of $L$, let:
:$\phi_p ... | From [[Frame Homomorphism Onto Two Induced by Completely Prime Filter]]:
:For every [[Definition:Completely Prime Filter|completely prime filter]] $p$ of $L$, $\phi_p : L \to \mathbf 2$ is a [[Definition:Frame Homomorphism|frame homomorphism]]
From [[Completely Prime Filter Induced by Frame Homomorphism Onto Two]]:
:... | Completely Prime Filter Induced by Frame Homomorphism Induced by Completely Prime Filter | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Frame_Homomorphism_Induced_by_Completely_Prime_Filter | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Frame_Homomorphism_Induced_by_Completely_Prime_Filter | [
"Completely Prime Filters",
"Two (Boolean Lattice)",
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Two (Boolean Lattice)",
"Definition:Completely Prime Filter",
"Definition:Frame Homomorphism",
"Definition:Frame Homomorphism",
"Definition:Preimage/Mapping/Element",
"Definition:Mapping",
"Definition:Completely Prime Filter",
"Definition:Completely P... | [
"Frame Homomorphism Onto Two Induced by Completely Prime Filter",
"Definition:Completely Prime Filter",
"Definition:Frame Homomorphism",
"Completely Prime Filter Induced by Frame Homomorphism Onto Two",
"Definition:Frame Homomorphism",
"Definition:Completely Prime Filter",
"Category:Completely Prime Fil... |
proofwiki-22355 | Frame Homomorphism Induced by Completely Prime Filter Induced by Frame Homomorphism | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the (Boolean Lattice) $\mathbf 2$.
For every frame homomorphism $\phi : L \to \mathbf 2$, let:
:$p_\phi = \map {\phi^{-1}} \top$
where
:$\map {\phi^{-1}} \top$ denotes the preimage of $\top \in \mathbf 2$ under ... | From Completely Prime Filter Induced by Frame Homomorphism Onto Two:
:For every frame homomorphism $\phi : L \to \mathbf 2$, $p_\phi = \map {\phi^{-1}} \top$ is a completely prime filter of $L$
From Frame Homomorphism Onto Two Induced by Completely Prime Filter:
:For every completely prime filter $p$ of $L$, $\phi_p : ... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]].
For every [[Definition:Frame Homomorphism|frame homomorphism]] $\phi : L \to \mathbf 2$, let:
:$p_\... | From [[Completely Prime Filter Induced by Frame Homomorphism Onto Two]]:
:For every [[Definition:Frame Homomorphism|frame homomorphism]] $\phi : L \to \mathbf 2$, $p_\phi = \map {\phi^{-1}} \top$ is a [[Definition:Completely Prime Filter|completely prime filter]] of $L$
From [[Frame Homomorphism Onto Two Induced by C... | Frame Homomorphism Induced by Completely Prime Filter Induced by Frame Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_Induced_by_Completely_Prime_Filter_Induced_by_Frame_Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_Induced_by_Completely_Prime_Filter_Induced_by_Frame_Homomorphism | [
"Completely Prime Filters",
"Two (Boolean Lattice)",
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Two (Boolean Lattice)",
"Definition:Frame Homomorphism",
"Definition:Preimage/Mapping/Element",
"Definition:Mapping",
"Definition:Completely Prime Filter",
"Definition:Completely Prime Filter",
"Definition:Frame Homomorphism",
"Definition:Frame Homomo... | [
"Completely Prime Filter Induced by Frame Homomorphism Onto Two",
"Definition:Frame Homomorphism",
"Definition:Completely Prime Filter",
"Frame Homomorphism Onto Two Induced by Completely Prime Filter",
"Definition:Completely Prime Filter",
"Definition:Frame Homomorphism",
"Equality of Mappings",
"Cat... |
proofwiki-22356 | Completely Prime Filter Induced by Meet Irreducible Induced by Completely Prime Filter | Let $\struct{L, \preceq}$ be a frame.
Let $\top$ denote the greatest element of $L$.
For every completely prime filter $p$ of $L$, let:
:$b_p = \bigvee \set{a \in L : a \notin p}$
where:
:$\bigvee \set{a \in L : a \notin p}$ denotes the supremum of the set $\set{a \in L : a \notin p}$
which is a meet irreducible elemen... | We have:
{{begin-eqn}}
{{eqn | l = p_{b_p}
| r = \set{a \in L : a \npreceq b_p}
| c = Definition of $p_{b_p}$
}}
{{eqn | r = \set{a \in L : a \npreceq \bigvee \set{a' \in L : a' \notin p} }
| c = Definition of $b_p$
}}
{{eqn | r = \set{a \in L : a \npreceq \bigvee \paren{L \setminus p} }
| c = {... | Let $\struct{L, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\top$ denote the [[Definition:Greatest Element|greatest element]] of $L$.
For every [[Definition:Completely Prime Filter|completely prime filter]] $p$ of $L$, let:
:$b_p = \bigvee \set{a \in L : a \notin p}$
where:
:$\bigvee \set{a \in ... | We have:
{{begin-eqn}}
{{eqn | l = p_{b_p}
| r = \set{a \in L : a \npreceq b_p}
| c = Definition of $p_{b_p}$
}}
{{eqn | r = \set{a \in L : a \npreceq \bigvee \set{a' \in L : a' \notin p} }
| c = Definition of $b_p$
}}
{{eqn | r = \set{a \in L : a \npreceq \bigvee \paren{L \setminus p} }
| c = {... | Completely Prime Filter Induced by Meet Irreducible Induced by Completely Prime Filter | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Meet_Irreducible_Induced_by_Completely_Prime_Filter | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Meet_Irreducible_Induced_by_Completely_Prime_Filter | [
"Completely Prime Filters",
"Meet Irreducible Elements"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Greatest Element",
"Definition:Completely Prime Filter",
"Definition:Supremum",
"Definition:Meet Irreducible Element",
"Definition:Equals",
"Definition:Meet Irreducible Element",
"Definition:Equals",
"Definition:Completely Prime Filter",
"Definition... | [
"Element of Completely Prime Filter iff Does not Precede Supremum of Relative Complement",
"Category:Completely Prime Filters",
"Category:Meet Irreducible Elements"
] |
proofwiki-22357 | Meet Irreducible Induced by Completely Prime Filter Induced by Meet Irreducible | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\top$ denote the greatest element of $L$.
For every meet-irreducible element $b$ of $L$ not equal to $\top$, let:
:$p_b = \set{a \in L : a \npreceq b}$
where
:$p_b$ is a completely prime filter of $L$.
For every completely prime filter $p$ of $L$, let:
:$b_p = \b... | We have:
{{begin-eqn}}
{{eqn | l = b_{p_b}
| r = \bigvee \set{a \in L : a \notin p_b}
| c = Definition of $b_{p_b}$
}}
{{eqn | r = \bigvee \set{a \in L : a \notin \set{x \in L : x \npreceq b} }
| c = Definition of $p_b$
}}
{{eqn | r = \bigvee \set{a \in L : a \in \set{x \in L : x \preceq b} }
... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\top$ denote the [[Definition:Greatest Element|greatest element]] of $L$.
For every [[Definition:Meet Irreducible Element|meet-irreducible element]] $b$ of $L$ not [[Definition:Equal|equal]] to $\top$, let:
:$p_b = \set{a ... | We have:
{{begin-eqn}}
{{eqn | l = b_{p_b}
| r = \bigvee \set{a \in L : a \notin p_b}
| c = Definition of $b_{p_b}$
}}
{{eqn | r = \bigvee \set{a \in L : a \notin \set{x \in L : x \npreceq b} }
| c = Definition of $p_b$
}}
{{eqn | r = \bigvee \set{a \in L : a \in \set{x \in L : x \preceq b} }
... | Meet Irreducible Induced by Completely Prime Filter Induced by Meet Irreducible | https://proofwiki.org/wiki/Meet_Irreducible_Induced_by_Completely_Prime_Filter_Induced_by_Meet_Irreducible | https://proofwiki.org/wiki/Meet_Irreducible_Induced_by_Completely_Prime_Filter_Induced_by_Meet_Irreducible | [
"Completely Prime Filters",
"Meet Irreducible Elements"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Greatest Element",
"Definition:Meet Irreducible Element",
"Definition:Equals",
"Definition:Completely Prime Filter",
"Definition:Completely Prime Filter",
"Definition:Supremum",
"Definition:Meet Irreducible Element",
"Definition:Equals",
"Definition... | [
"Supremum of Lower Closure of Element",
"Category:Completely Prime Filters",
"Category:Meet Irreducible Elements"
] |
proofwiki-22358 | Existence of Schur Decomposition for Square Matrix | Let $\mathbf A$ be a square matrix.
Then there exists a '''Schur decomposition''' for $\mathbf A$. | Proof by induction.
For all $n \in \N_{>0}$, let $\map P n$ represent the proposition:
:All square matrices of order $n$ are unitarily similar to an upper triangular matrix.
That is, they have a Schur decomposition.
{{expand|We need a note somewhere on Definition:Unitary Transformation probably to mention exactly what ... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]].
Then there exists a '''[[Definition:Schur Decomposition|Schur decomposition]]''' for $\mathbf A$. | Proof by [[Definition:Principle of Mathematical Induction|induction]].
For all $n \in \N_{>0}$, let $\map P n$ represent the [[Definition:Proposition|proposition]]:
:All [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Matrix|order]] $n$ are [[Definition:Unitary Transformation|unitarily similar]]... | Existence of Schur Decomposition for Square Matrix | https://proofwiki.org/wiki/Existence_of_Schur_Decomposition_for_Square_Matrix | https://proofwiki.org/wiki/Existence_of_Schur_Decomposition_for_Square_Matrix | [
"Schur Decompositions",
"Square Matrices",
"Proofs by Induction"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Schur Decomposition"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Order",
"Definition:Unitary Transformation",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Schur Decomposition",
"Definition:Unitary Transformation",
"Definiti... |
proofwiki-22359 | Secant Method can be derived from Newton-Raphson Method | Let $f: \R \to \R$ be a real function which has a root which is to be found.
The '''secant method''' can be derived from the '''Newton-Raphson method'''. | {{Recall|Secant Method}}
{{:Definition:Secant Method}}
{{Recall|Newton-Raphson Method}}
{{:Definition:Newton-Raphson Method}}
We note that:
:$\map {f'} {x_n} \approx \dfrac {\map f x - \map f {x_{n - 1} } } {x_n - x_{n - 1} }$
The result follows by substituting for $\map {f'} {x_n}$ in the definition of the Newton-Raph... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which has a [[Definition:Root of Function|root]] which is to be found.
The '''[[Definition:Secant Method|secant method]]''' can be derived from the '''[[Definition:Newton-Raphson Method|Newton-Raphson method]]'''. | {{Recall|Secant Method}}
{{:Definition:Secant Method}}
{{Recall|Newton-Raphson Method}}
{{:Definition:Newton-Raphson Method}}
We note that:
:$\map {f'} {x_n} \approx \dfrac {\map f x - \map f {x_{n - 1} } } {x_n - x_{n - 1} }$
The result follows by substituting for $\map {f'} {x_n}$ in the definition of the [[Defini... | Secant Method can be derived from Newton-Raphson Method | https://proofwiki.org/wiki/Secant_Method_can_be_derived_from_Newton-Raphson_Method | https://proofwiki.org/wiki/Secant_Method_can_be_derived_from_Newton-Raphson_Method | [
"Secant Method",
"Newton-Raphson Method"
] | [
"Definition:Real Function",
"Definition:Root of Mapping",
"Definition:Secant Method",
"Definition:Newton-Raphson Method"
] | [
"Definition:Newton-Raphson Method"
] |
proofwiki-22360 | Complement of Countable Set in Topological Space where Open Neighborhood is Uncountable is Everywhere Dense | Let $\struct {X, \tau}$ be a topological space such that:
:every open set $U \in \tau$ is uncountable.
Let $A \subseteq X$ be countable.
Then $X \setminus A$ is everywhere dense in $\struct {X, \tau}$. | Let $a \in X$.
Let $U$ be an open neighborhood of $a$ in $\struct {X, \tau}$.
Then $U$ is uncountable by hypothesis.
From Uncountable Set less Countable Set is Uncountable, $U \cap \paren {X \setminus A}$ is uncountable.
In particular, $U \cap \paren {X \setminus A} \ne \O$.
Hence we have $a \in \paren {X \setminus A... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]] such that:
:every [[Definition:Open Set|open set]] $U \in \tau$ is [[Definition:Uncountable Set|uncountable]].
Let $A \subseteq X$ be [[Definition:Countable Set|countable]].
Then $X \setminus A$ is [[Definition:Everywhere Dense|everywher... | Let $a \in X$.
Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of $a$ in $\struct {X, \tau}$.
Then $U$ is [[Definition:Uncountable Set|uncountable]] by hypothesis.
From [[Uncountable Set less Countable Set is Uncountable]], $U \cap \paren {X \setminus A}$ is [[Definition:Uncountable Set|uncountabl... | Complement of Countable Set in Topological Space where Open Neighborhood is Uncountable is Everywhere Dense | https://proofwiki.org/wiki/Complement_of_Countable_Set_in_Topological_Space_where_Open_Neighborhood_is_Uncountable_is_Everywhere_Dense | https://proofwiki.org/wiki/Complement_of_Countable_Set_in_Topological_Space_where_Open_Neighborhood_is_Uncountable_is_Everywhere_Dense | [
"Complement of Countable Set in Topological Space where Open Neighborhood is Uncountable is Everywhere Dense",
"Topological Spaces",
"Complement of Countable Set in Topological Space where Open Neighborhood is Uncountable is Everywhere Dense"
] | [
"Definition:Topological Space",
"Definition:Open Set",
"Definition:Uncountable/Set",
"Definition:Countable Set",
"Definition:Everywhere Dense"
] | [
"Definition:Open Neighborhood",
"Definition:Uncountable/Set",
"Uncountable Set less Countable Set is Uncountable",
"Definition:Uncountable/Set",
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Category:Topological Spaces",
"Category:Complement of Countable Set in Topological Space whe... |
proofwiki-22361 | Ramanujan's Arctangent Sum | :$\ds \sum_{k \mathop = 0}^{r - 1} \map \arctan {\dfrac 2 {\paren {n + 2 k + 1}^2} } = \map \arctan {\dfrac {2 r} {\paren {n^2 + 2 n r + 1} } }$ | Recall the Difference of Arctangents:
{{:Difference of Arctangents}}
We first observe:
{{begin-eqn}}
{{eqn | l = \map \arctan {\frac 1 {\paren {n + 2 k} } } - \map \arctan {\frac 1 {\paren {n + 2 k + 2} } }
| r = \map \arctan {\dfrac {\dfrac 1 {\paren {n + 2 k } } - \dfrac 1 {\paren {n + 2 k + 2} } } {1 + \dfrac ... | :$\ds \sum_{k \mathop = 0}^{r - 1} \map \arctan {\dfrac 2 {\paren {n + 2 k + 1}^2} } = \map \arctan {\dfrac {2 r} {\paren {n^2 + 2 n r + 1} } }$ | Recall the [[Difference of Arctangents]]:
{{:Difference of Arctangents}}
We first observe:
{{begin-eqn}}
{{eqn | l = \map \arctan {\frac 1 {\paren {n + 2 k} } } - \map \arctan {\frac 1 {\paren {n + 2 k + 2} } }
| r = \map \arctan {\dfrac {\dfrac 1 {\paren {n + 2 k } } - \dfrac 1 {\paren {n + 2 k + 2} } } {1 + \d... | Ramanujan's Arctangent Sum | https://proofwiki.org/wiki/Ramanujan's_Arctangent_Sum | https://proofwiki.org/wiki/Ramanujan's_Arctangent_Sum | [
"Ramanujan's Arctangent Sum",
"Arctangent Function"
] | [] | [
"Difference of Arctangents",
"Difference of Arctangents"
] |
proofwiki-22362 | Real-Valued Continuous Function with Compact Support is Uniformly Continuous | Let $f : \R \to \R$ be a continuous function such that:
:the support $\map \supp f$ is compact.
Then $f$ is uniformly continuous. | Since $\map \supp f$ is compact, we can take $\alpha > 0$ such that:
:$\map \supp f \subseteq \closedint {-\alpha} \alpha$
and $\map f x = 0$ for $\cmod x \ge \alpha$.
From the Heine-Cantor Theorem, $f$ is uniformly continuous on $\closedint {-\alpha - 1} {\alpha + 1}$.
Let $\epsilon > 0$.
Then there exists $\delta' >... | Let $f : \R \to \R$ be a [[Definition:Continuous Function|continuous function]] such that:
:the [[Definition:Support of Continuous Mapping|support]] $\map \supp f$ is [[Definition:Compact Topological Space|compact]].
Then $f$ is [[Definition:Uniformly Continuous Real Function|uniformly continuous]]. | Since $\map \supp f$ is [[Definition:Compact Topological Space|compact]], we can take $\alpha > 0$ such that:
:$\map \supp f \subseteq \closedint {-\alpha} \alpha$
and $\map f x = 0$ for $\cmod x \ge \alpha$.
From the [[Heine-Cantor Theorem]], $f$ is [[Definition:Uniformly Continuous Real Function|uniformly continuou... | Real-Valued Continuous Function with Compact Support is Uniformly Continuous | https://proofwiki.org/wiki/Real-Valued_Continuous_Function_with_Compact_Support_is_Uniformly_Continuous | https://proofwiki.org/wiki/Real-Valued_Continuous_Function_with_Compact_Support_is_Uniformly_Continuous | [
"Uniformly Continuous Real Functions"
] | [
"Definition:Continuous Function",
"Definition:Support of Continuous Mapping",
"Definition:Compact Topological Space",
"Definition:Uniform Continuity/Real Function"
] | [
"Definition:Compact Topological Space",
"Heine-Cantor Theorem",
"Definition:Uniform Continuity/Real Function",
"Definition:Uniform Continuity/Real Function",
"Category:Uniformly Continuous Real Functions"
] |
proofwiki-22363 | Shannon's Theorem | Let $T$ be a transmission channel subject to random errors.
Then there exists an effective error-correcting code for $T$. | {{ProofWanted}}
{{Namedfor|Claude Elwood Shannon|cat = Shannon}} | Let $T$ be a [[Definition:Transmission Channel|transmission channel]] subject to [[Definition:Random Error|random errors]].
Then there exists an effective [[Definition:Error-Correcting Code|error-correcting code]] for $T$. | {{ProofWanted}}
{{Namedfor|Claude Elwood Shannon|cat = Shannon}} | Shannon's Theorem | https://proofwiki.org/wiki/Shannon's_Theorem | https://proofwiki.org/wiki/Shannon's_Theorem | [
"Information Theory"
] | [
"Definition:Transmission Channel",
"Definition:Random Error",
"Definition:Error-Correcting Code"
] | [] |
proofwiki-22364 | Completely Prime Ideal is Dual of Completely Prime Filter | Let $\struct{L, \preceq}$ be a complete lattice.
Let $K \subseteq L$.
The following are dual statements:
:$K$ is a completely prime filter of $\struct {L, \preceq}$
:$K$ is a completely prime ideal of $\struct {L, \preceq}$ | By definition of completely prime filter:
:$K$ is a completely prime filter of $\struct {L, \preceq}$
{{iff}}:
:$(1)\quad K$ is a proper filter
:$(2)\quad \forall A \subseteq L: \paren{\wedge A \in K \implies A \cap K \ne \O}$
where $\wedge A$ denotes the infimum of $A$
By the duality principle, the dual of this statem... | Let $\struct{L, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $K \subseteq L$.
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$K$ is a [[Definition:Completely Prime Filter|completely prime filter]] of $\struct {L, \preceq}$
:$K$ is a [[Definition:Completely ... | By definition of [[Definition:Completely Prime Filter|completely prime filter]]:
:$K$ is a [[Definition:Completely Prime Filter|completely prime filter]] of $\struct {L, \preceq}$
{{iff}}:
:$(1)\quad K$ is a [[Definition:Proper Filter|proper filter]]
:$(2)\quad \forall A \subseteq L: \paren{\wedge A \in K \implies A \c... | Completely Prime Ideal is Dual of Completely Prime Filter | https://proofwiki.org/wiki/Completely_Prime_Ideal_is_Dual_of_Completely_Prime_Filter | https://proofwiki.org/wiki/Completely_Prime_Ideal_is_Dual_of_Completely_Prime_Filter | [
"Dual Pairs (Order Theory)",
"Completely Prime Filters",
"Completely Prime Ideals"
] | [
"Definition:Complete Lattice",
"Definition:Dual Statement (Order Theory)",
"Definition:Completely Prime Filter",
"Definition:Completely Prime Ideal"
] | [
"Definition:Completely Prime Filter",
"Definition:Completely Prime Filter",
"Definition:Filter/Proper Filter",
"Definition:Infimum of Set",
"Duality Principle (Order Theory)",
"Definition:Dual Statement (Order Theory)",
"Definition:Ideal (Order Theory)/Proper Ideal",
"Definition:Supremum of Set",
"D... |
proofwiki-22365 | Ideal is Dual of Filter (Order Theory) | Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be a subset of $S$.
The following are dual statements:
:$T$ is a filter of $\struct {S, \preceq}$
:$T$ is an ideal of $\struct {S, \preceq}$ | By definition of filter:
:$T$ is a filter of $\struct {S, \preceq}$
{{iff}}:
{{begin-axiom}}
{{axiom | n = 1
| m = T \ne \O
}}
{{axiom | n = 2
| m = x, y \in T \implies \exists z \in T: z \preccurlyeq x, z \preccurlyeq y
}}
{{axiom | n = 3
| m = \forall x \in T: \forall y \in S: x \preccurlyeq y... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$T$ is a [[Definition:Filter|filter]] of $\struct {S, \preceq}$
:$T$ is an [[Definition:Ideal (Order... | By definition of [[Definition:Filter|filter]]:
:$T$ is a [[Definition:Filter|filter]] of $\struct {S, \preceq}$
{{iff}}:
{{begin-axiom}}
{{axiom | n = 1
| m = T \ne \O
}}
{{axiom | n = 2
| m = x, y \in T \implies \exists z \in T: z \preccurlyeq x, z \preccurlyeq y
}}
{{axiom | n = 3
| m = \foral... | Ideal is Dual of Filter (Order Theory) | https://proofwiki.org/wiki/Ideal_is_Dual_of_Filter_(Order_Theory) | https://proofwiki.org/wiki/Ideal_is_Dual_of_Filter_(Order_Theory) | [
"Dual Pairs (Order Theory)",
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Dual Statement (Order Theory)",
"Definition:Filter",
"Definition:Ideal (Order Theory)"
] | [
"Definition:Filter",
"Definition:Filter",
"Duality Principle (Order Theory)",
"Definition:Dual Statement (Order Theory)",
"Definition:Ideal (Order Theory)",
"Definition:Ideal (Order Theory)",
"Category:Dual Pairs (Order Theory)",
"Category:Order Theory"
] |
proofwiki-22366 | Characterization of Completely Prime Ideal in Complete Lattice | Let $\struct{L, \vee, \wedge, \preceq}$ be a complete lattice.
Let $I \subseteq L$.
Then:
:$I$ is a completely prime ideal
{{iff}}
:$(1)\quad\forall A \subseteq L : \bigwedge A \in I \iff \paren{\exists a \in A : a \in I}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigvee A \in I \iff \paren{\forall a \in A : a \in I... | This is the dual statement of Characterization of Completely Prime Filter in Complete Lattice by Dual Pairs (Order Theory).
The result follows from the Duality Principle.
{{qed}}
Category:Complete Lattices
Category:Completely Prime Ideals
kljmrxbby0m3t3jw55dcp3nczxjatig | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $I \subseteq L$.
Then:
:$I$ is a [[Definition:Completely Prime Filter|completely prime ideal]]
{{iff}}
:$(1)\quad\forall A \subseteq L : \bigwedge A \in I \iff \paren{\exists a \in A : a \in I}$
:$(2)\quad\forall $ [[... | This is the [[Definition:Dual Statement (Order Theory)|dual statement]] of [[Characterization of Completely Prime Filter in Complete Lattice]] by [[Dual Pairs (Order Theory)]].
The result follows from the [[Duality Principle (Order Theory)|Duality Principle]].
{{qed}}
[[Category:Complete Lattices]]
[[Category:Complet... | Characterization of Completely Prime Ideal in Complete Lattice | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Ideal_in_Complete_Lattice | https://proofwiki.org/wiki/Characterization_of_Completely_Prime_Ideal_in_Complete_Lattice | [
"Complete Lattices",
"Completely Prime Ideals"
] | [
"Definition:Complete Lattice",
"Definition:Completely Prime Filter",
"Definition:Finite",
"Definition:Infimum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Dual Statement (Order Theory)",
"Characterization of Completely Prime Filter in Complete Lattice",
"Dual Pairs (Order Theory)",
"Duality Principle (Order Theory)",
"Category:Complete Lattices",
"Category:Completely Prime Ideals"
] |
proofwiki-22367 | Simpson's Rule/Repeated | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:
:$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$
where $n$ is even.
Then the definite integral of $f$ {... | {{ProofWanted|Graphical approach based on approximating the area under the curve as a series of parabolas.}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a [[Definition:Normal Subdivision|normal subdivision]] of $\closedint a... | {{ProofWanted|Graphical approach based on approximating the area under the curve as a series of parabolas.}} | Simpson's Rule/Repeated | https://proofwiki.org/wiki/Simpson's_Rule/Repeated | https://proofwiki.org/wiki/Simpson's_Rule/Repeated | [
"Simpson's Rule"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval/Normal Subdivision",
"Definition:Even Integer",
"Definition:Definite Integral",
"Definition:Approximation",
"Simpson's Rule/Repeated"
] | [] |
proofwiki-22368 | Element of Completely Prime Filter iff Does not Precede Supremum of Relative Complement | Let $\struct{L, \preceq}$ be a complete lattice.
Let $p$ be a completely prime filter $p$ of $L$.
Let $a \in L$.
Then:
:$a \in p$
{{iff}}:
:$a \npreceq \bigvee \paren{L \setminus p}$
where:
:$L \setminus p$ denotes the relative complement of $p$ in $L$
:$\bigvee \paren{L \setminus p}$ denotes the supremum of $L \setmin... | === Necessary Condition ===
We show the contrapositive statement:
:$a \preceq \bigvee \paren{L \setminus p} \implies a \notin p$
Let:
:$a \preceq \bigvee \paren{L \setminus p}$
By definition of completely prime filter:
:$\bigvee \paren{L \setminus p} \in L \setminus p$
By definition of relative complement:
:$\bigvee \... | Let $\struct{L, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $p$ be a [[Definition:Completely Prime Filter|completely prime filter]] $p$ of $L$.
Let $a \in L$.
Then:
:$a \in p$
{{iff}}:
:$a \npreceq \bigvee \paren{L \setminus p}$
where:
:$L \setminus p$ denotes the [[Definition:Relative Comp... | === Necessary Condition ===
We show the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$a \preceq \bigvee \paren{L \setminus p} \implies a \notin p$
Let:
:$a \preceq \bigvee \paren{L \setminus p}$
By definition of [[Definition:Completely Prime Filter|completely prime filter]]:
:$\bigvee \paren{... | Element of Completely Prime Filter iff Does not Precede Supremum of Relative Complement | https://proofwiki.org/wiki/Element_of_Completely_Prime_Filter_iff_Does_not_Precede_Supremum_of_Relative_Complement | https://proofwiki.org/wiki/Element_of_Completely_Prime_Filter_iff_Does_not_Precede_Supremum_of_Relative_Complement | [
"Completely Prime Filters"
] | [
"Definition:Complete Lattice",
"Definition:Completely Prime Filter",
"Definition:Relative Complement",
"Definition:Supremum of Set"
] | [
"Definition:Contrapositive Statement",
"Definition:Completely Prime Filter",
"Definition:Relative Complement",
"Definition:Filter",
"Definition:Relative Complement"
] |
proofwiki-22369 | Bounded Right-Continuous Increasing Function Vanishing at Infimum is Distribution Function of Finite Borel Measure | Let $K$ be a closed interval.
Let $F : K \to \R$ be a bounded right-continuous increasing function such that:
:$\ds \lim_{x \mathop \to \inf K} \map F x = 0$
and:
:$\ds \lim_{x \mathop \to \sup K} \map F x = M \in \hointr 0 \infty$
Then there exists a finite Borel measure $\mu$ on $K$ such that the distribution functi... | If $M = 0$, then we can take $\mu = 0$.
Hence suppose that $M \ne 0$.
Note now that from Limit of Monotone Real Function, we have:
:$\ds M = \sup_{x \mathop \in K} \map F x$
For $t \in \openint 0 M$, define:
:$\map Q t = \inf \set {x \in K : \map F x \ge t}$
We first show that $\map Q t \in \openint {\inf K} {\sup K}... | Let $K$ be a [[Definition:Closed Interval|closed interval]].
Let $F : K \to \R$ be a [[Definition:Bounded Mapping|bounded]] [[Definition:Right-Continuous Real Function|right-continuous]] [[Definition:Increasing Function|increasing function]] such that:
:$\ds \lim_{x \mathop \to \inf K} \map F x = 0$
and:
:$\ds \lim_{x... | If $M = 0$, then we can take $\mu = 0$.
Hence suppose that $M \ne 0$.
Note now that from [[Limit of Monotone Real Function]], we have:
:$\ds M = \sup_{x \mathop \in K} \map F x$
For $t \in \openint 0 M$, define:
:$\map Q t = \inf \set {x \in K : \map F x \ge t}$
We first show that $\map Q t \in \openint {\inf K} ... | Bounded Right-Continuous Increasing Function Vanishing at Infimum is Distribution Function of Finite Borel Measure | https://proofwiki.org/wiki/Bounded_Right-Continuous_Increasing_Function_Vanishing_at_Infimum_is_Distribution_Function_of_Finite_Borel_Measure | https://proofwiki.org/wiki/Bounded_Right-Continuous_Increasing_Function_Vanishing_at_Infimum_is_Distribution_Function_of_Finite_Borel_Measure | [
"Bounded Right-Continuous Increasing Function Vanishing at Infimum is Distribution Function of Finite Borel Measure",
"Distribution Function of Finite Borel Measure"
] | [
"Definition:Interval/Ordered Set/Closed",
"Definition:Bounded Mapping",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Increasing/Real Function",
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Distribution Function of Finite Borel Measure"
] | [
"Limit of Monotone Real Function",
"Definition:Increasing/Real Function",
"Definition:Interval/Ordered Set/Closed",
"Definition:Measurable Function",
"Infimum of Subset",
"Definition:Increasing/Real Function",
"Monotone Real Function is Measurable",
"Definition:Measurable Function",
"Definition:Lebe... |
proofwiki-22370 | Monotone Real Function is Measurable | Let $I \subseteq \R$ be an open interval.
Let $\map \BB I$ and $\map \BB \R$ be the Borel $\sigma$-algebras of $I$ and $\R$ respectively.
Let $F : \R \to \R$ be a monotone function.
Then $F$ is $\map \BB I/\map \BB \R$-measurable. | We first assume that $F$ is increasing.
From Borel Sigma-Algebra of Subset is Trace Sigma-Algebra, $\map \BB I$ is the trace $\sigma$-algebra of $I$ in $\map \BB \R$.
From Characterization of Measurable Functions, it is enough to show that:
:$J_t = \set {x \in I : \map F x \le t} \in \map \BB I$ for each $t \in \R$.
... | Let $I \subseteq \R$ be an [[Definition:Open Interval|open interval]].
Let $\map \BB I$ and $\map \BB \R$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebras]] of $I$ and $\R$ respectively.
Let $F : \R \to \R$ be a [[Definition:Monotone Real Function|monotone function]].
Then $F$ is [[Definition:Measura... | We first assume that $F$ is [[Definition:Increasing Function|increasing]].
From [[Borel Sigma-Algebra of Subset is Trace Sigma-Algebra]], $\map \BB I$ is the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra]] of $I$ in $\map \BB \R$.
From [[Characterization of Measurable Functions]], it is enough to show that... | Monotone Real Function is Measurable | https://proofwiki.org/wiki/Monotone_Real_Function_is_Measurable | https://proofwiki.org/wiki/Monotone_Real_Function_is_Measurable | [
"Monotone Real Functions",
"Measurable Functions",
"Monotone Real Functions"
] | [
"Definition:Interval/Ordered Set/Open",
"Definition:Borel Sigma-Algebra",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Measurable Mapping"
] | [
"Definition:Increasing/Real Function",
"Borel Sigma-Algebra of Subset is Trace Sigma-Algebra",
"Definition:Trace Sigma-Algebra",
"Characterization of Measurable Functions",
"Measure of Interval is Length",
"Definition:Trace Sigma-Algebra",
"Definition:Real Interval",
"Definition:Increasing/Real Functi... |
proofwiki-22371 | Right Limit Function is Right-Continuous | Let $I$ be an open interval.
Let $f : I \to \R$ be a real function such that:
:for each $x \in I$, the right limit:
::$\ds \lim_{y \mathop \to x^+} \map f y$
:exists.
Define $f_\leftarrow : I \to \R$ by:
:$\ds \map {f_\leftarrow} x = \lim_{y \mathop \to x^+} \map f y$
for each $x \in I$.
Then $f_\leftarrow$ is right-c... | Let $\epsilon > 0$.
Let $x \in I$.
By the definition of the right limit at $x$, there exists $\delta > 0$ such that for all $y \in I$ satisfying $x < y < x + \delta$, we have:
:$\ds \size {\map f y - \map {f_\leftarrow} x} < \frac \epsilon 3$
Take $y \in I$ such that $x < y < x + \delta$.
Let $\sequence {x_n}_{n \in... | Let $I$ be an [[Definition:Open Interval|open interval]].
Let $f : I \to \R$ be a [[Definition:Real Function|real function]] such that:
:for each $x \in I$, the [[Definition:Limit from Right|right limit]]:
::$\ds \lim_{y \mathop \to x^+} \map f y$
:exists.
Define $f_\leftarrow : I \to \R$ by:
:$\ds \map {f_\leftarro... | Let $\epsilon > 0$.
Let $x \in I$.
By the definition of the [[Definition:Limit from Right|right limit]] at $x$, there exists $\delta > 0$ such that for all $y \in I$ satisfying $x < y < x + \delta$, we have:
:$\ds \size {\map f y - \map {f_\leftarrow} x} < \frac \epsilon 3$
Take $y \in I$ such that $x < y < x + \d... | Right Limit Function is Right-Continuous | https://proofwiki.org/wiki/Right_Limit_Function_is_Right-Continuous | https://proofwiki.org/wiki/Right_Limit_Function_is_Right-Continuous | [
"Right-Continuous Functions"
] | [
"Definition:Interval/Ordered Set/Open",
"Definition:Real Function",
"Definition:Limit of Real Function/Right",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Definition:Limit of Real Function/Right",
"Definition:Sequence",
"Definition:Sequence",
"Limit of Function by Convergent Sequences/Corollary",
"Squeeze Theorem",
"Definition:Continuous Real Function/Right-Continuous",
"Category:Right-Continuous Functions"
] |
proofwiki-22372 | Derived Subgroup of Abelian Group is Trivial | Let $G$ be a abelian group with identity $e$.
Let $G'$ be the derived subgroup of $G$.
Then $G'$ is the trivial group $\set e$. | For $g, h \in G$, let $\sqbrk {g, h}$ denote the commutator of $g$ and $h$:
:$\sqbrk {g, h} := g^{-1} \circ h^{-1} \circ g \circ h$
Let $C$ be the set of the commutators of $G$:
:$C = \set {\sqbrk {g, h}: g, h \in G}$
From Commutators are Identity iff Group is Abelian, $C = \set e$.
From Trivial Subgroup is Subgroup, $... | Let $G$ be a [[Definition:Abelian Group|abelian group]] with identity $e$.
Let $G'$ be the [[Definition:Derived Subgroup|derived subgroup]] of $G$.
Then $G'$ is the [[Definition:Trivial Group|trivial group]] $\set e$. | For $g, h \in G$, let $\sqbrk {g, h}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $g$ and $h$:
:$\sqbrk {g, h} := g^{-1} \circ h^{-1} \circ g \circ h$
Let $C$ be the [[Definition:Set|set]] of the [[Definition:Commutator of Group Elements|commutators]] of $G$:
:$C = \set {\sqbrk {g, h}: g, h \i... | Derived Subgroup of Abelian Group is Trivial | https://proofwiki.org/wiki/Derived_Subgroup_of_Abelian_Group_is_Trivial | https://proofwiki.org/wiki/Derived_Subgroup_of_Abelian_Group_is_Trivial | [
"Derived Subgroups",
"Abelian Groups",
"Trivial Group"
] | [
"Definition:Abelian Group",
"Definition:Derived Subgroup",
"Definition:Trivial Group"
] | [
"Definition:Commutator/Group",
"Definition:Set",
"Definition:Commutator/Group",
"Commutators are Identity iff Group is Abelian",
"Trivial Subgroup is Subgroup",
"Definition:Subgroup",
"Subgroup Generated by Subgroup",
"Definition:Generated Subgroup",
"Definition:Derived Subgroup"
] |
proofwiki-22373 | Biholomorphic Function from Open Unit Disk to Right Half-Plane | Let $\mathbb D = \set {z : \cmod z < 1}$ be the open unit disk.
Let $\mathbb H_r = \set {z : \map \Re z > 0}$ be the right half-plane.
Define $f : \mathbb D \to \C$ by:
:$\ds \map f z = \frac {1 - z} {1 + z}$
for each $z \in \mathbb D$.
Then $f$ is a biholomorphic function $\mathbb D \to \mathbb H_r$. | From Möbius Transformation is Bijection, defining the extended map $f_\ast : \overline \C \to \overline \C$ by:
:<nowiki>$\map {f_\ast} z = \begin {cases} \dfrac {1 - z} {1 + z} & : z \ne -1 \\
\infty & : z = -1 \\
-1 & : z = \infty \end{cases}$</nowiki>
we obtain a bijection.
Hence $f : \mathbb D \to \C$ is an injecti... | Let $\mathbb D = \set {z : \cmod z < 1}$ be the [[Definition:Open Ball|open]] [[Definition:Unit Disk|unit disk]].
Let $\mathbb H_r = \set {z : \map \Re z > 0}$ be the [[Definition:Right Half-Plane|right half-plane]].
Define $f : \mathbb D \to \C$ by:
:$\ds \map f z = \frac {1 - z} {1 + z}$
for each $z \in \mathbb D$... | From [[Möbius Transformation is Bijection]], defining the [[Definition:Extension of Mapping|extended map]] $f_\ast : \overline \C \to \overline \C$ by:
:<nowiki>$\map {f_\ast} z = \begin {cases} \dfrac {1 - z} {1 + z} & : z \ne -1 \\
\infty & : z = -1 \\
-1 & : z = \infty \end{cases}$</nowiki>
we obtain a [[Definition:... | Biholomorphic Function from Open Unit Disk to Right Half-Plane | https://proofwiki.org/wiki/Biholomorphic_Function_from_Open_Unit_Disk_to_Right_Half-Plane | https://proofwiki.org/wiki/Biholomorphic_Function_from_Open_Unit_Disk_to_Right_Half-Plane | [
"Biholomorphic Functions",
"Möbius Transformations"
] | [
"Definition:Open Ball",
"Definition:Unit Disk",
"Definition:Half-Plane/Right",
"Definition:Biholomorphic Function"
] | [
"Möbius Transformation is Bijection",
"Definition:Extension of Mapping",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Surjection",
"Sum of Complex Conjugates",
"Product of Complex Number with Conjugate",
"Difference of Complex Number with Conjugate",
"Produc... |
proofwiki-22374 | Vague Convergence of Uniformly Bounded Sequence of Finite Borel Measures implies Convergence of Integrals of Bounded Continuous Functions Vanishing at Infinity | Let $M > 0$.
Let $\mu$ be a finite Borel measure on $\R$ with $\map \mu \R \le M$.
Let $\sequence {\mu_n}_{n \mathop \in \N}$ be a sequence of finite Borel measures converging vaguely to $\mu$ such that:
:$\map {\mu_n} \R \le M$ for each $n \in \N$.
Let $f : \R \to \R$ be continuous such that:
:$\ds \lim_{\size x \mat... | Let $\epsilon > 0$.
Since:
:$\ds \lim_{\size x \mathop \to \infty} \map f x = 0$
there exists $K > 0$ such that:
:$\ds \size {\map f x} < \frac \epsilon 2$ for $\size x \ge K$.
Define $f_1 : \R \to \R$ by:
:$\ds \map {f_1} x = \begin {cases} \map f x & : x \in \closedint {-K} K \\ \paren {x - K + 1} \map f K & : x \in ... | Let $M > 0$.
Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measure]] on $\R$ with $\map \mu \R \le M$.
Let $\sequence {\mu_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Finite Measure|finite]] [[Definition:Borel Measure|Borel measures]] [[Definiti... | Let $\epsilon > 0$.
Since:
:$\ds \lim_{\size x \mathop \to \infty} \map f x = 0$
there exists $K > 0$ such that:
:$\ds \size {\map f x} < \frac \epsilon 2$ for $\size x \ge K$.
Define $f_1 : \R \to \R$ by:
:$\ds \map {f_1} x = \begin {cases} \map f x & : x \in \closedint {-K} K \\ \paren {x - K + 1} \map f K & : x \i... | Vague Convergence of Uniformly Bounded Sequence of Finite Borel Measures implies Convergence of Integrals of Bounded Continuous Functions Vanishing at Infinity | https://proofwiki.org/wiki/Vague_Convergence_of_Uniformly_Bounded_Sequence_of_Finite_Borel_Measures_implies_Convergence_of_Integrals_of_Bounded_Continuous_Functions_Vanishing_at_Infinity | https://proofwiki.org/wiki/Vague_Convergence_of_Uniformly_Bounded_Sequence_of_Finite_Borel_Measures_implies_Convergence_of_Integrals_of_Bounded_Continuous_Functions_Vanishing_at_Infinity | [
"Vague Convergence of Borel Measures"
] | [
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Sequence",
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Vague Convergence of Borel Measures",
"Definition:Continuous Function"
] | [
"Definition:Compact Topological Space",
"Definition:Support of Continuous Mapping",
"Definition:Continuous Function",
"Definition:Vague Convergence of Borel Measures",
"Triangle Inequality",
"Triangle Inequality",
"Integral of Integrable Function is Additive",
"Triangle Inequality",
"Triangle Inequa... |
proofwiki-22375 | Derived Subgroup is Trivial iff Group is Abelian | Let $G$ be a group with identity $e$.
Let $G'$ be the derived subgroup of $G$.
Then:
:$G'$ is the trivial group $\set e$
{{iff}}:
:$G$ is abelian. | === Sufficient Condition ===
This is proved in Derived Subgroup of Abelian Group is Trivial.
{{qed|lemma}} | Let $G$ be a [[Definition:Group|group]] with identity $e$.
Let $G'$ be the [[Definition:Derived Subgroup|derived subgroup]] of $G$.
Then:
:$G'$ is the [[Definition:Trivial Group|trivial group]] $\set e$
{{iff}}:
:$G$ is [[Definition:Abelian Group|abelian]]. | === Sufficient Condition ===
This is proved in [[Derived Subgroup of Abelian Group is Trivial]].
{{qed|lemma}} | Derived Subgroup is Trivial iff Group is Abelian | https://proofwiki.org/wiki/Derived_Subgroup_is_Trivial_iff_Group_is_Abelian | https://proofwiki.org/wiki/Derived_Subgroup_is_Trivial_iff_Group_is_Abelian | [
"Derived Subgroups",
"Abelian Groups",
"Trivial Group"
] | [
"Definition:Group",
"Definition:Derived Subgroup",
"Definition:Trivial Group",
"Definition:Abelian Group"
] | [
"Derived Subgroup of Abelian Group is Trivial"
] |
proofwiki-22376 | Skew Lines Cannot Exist in Euclidean Plane | Let $\PP$ be a Euclidean plane.
Let $\LL_1$ and $\LL_2$ be straight lines embedded in $\PP$.
Then it is not possible for $\LL_1$ and $\LL_2$ to be '''skew''' | {{Recall|Skew Lines}}
{{:Definition:Skew Lines}}
The proof then follows directly from {{EuclidPostulateLink|Fifth}}:
{{:Axiom:Euclid's Fifth Postulate}}
That is, if $\LL_1$ and $\LL_2$ do not intersect, they are parallel.
{{qed}} | Let $\PP$ be a [[Definition:Euclidean Plane|Euclidean plane]].
Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in $\PP$.
Then it is not possible for $\LL_1$ and $\LL_2$ to be '''[[Definition:Skew Lines|skew]]''' | {{Recall|Skew Lines}}
{{:Definition:Skew Lines}}
The proof then follows directly from {{EuclidPostulateLink|Fifth}}:
{{:Axiom:Euclid's Fifth Postulate}}
That is, if $\LL_1$ and $\LL_2$ do not [[Definition:Intersection (Geometry)|intersect]], they are [[Definition:Parallel Lines|parallel]].
{{qed}} | Skew Lines Cannot Exist in Euclidean Plane | https://proofwiki.org/wiki/Skew_Lines_Cannot_Exist_in_Euclidean_Plane | https://proofwiki.org/wiki/Skew_Lines_Cannot_Exist_in_Euclidean_Plane | [
"Skew Lines"
] | [
"Definition:Euclidean Plane",
"Definition:Line/Straight Line",
"Definition:Skew Lines"
] | [
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines"
] |
proofwiki-22377 | Intersection of Plane with Sphere is Great Circle iff Passing through Center | Let $S$ be a sphere.
Let $P$ be a plane which intersects $S$ but is not tangent to $S$.
Let $C$ be the circle formed by the intersection between $P$ and $S$.
Then $C$ is a '''great circle''' of $S$ {{iff}} $P$ passes through the center of $S$.
Otherwise $C$ is a '''small circle''' of $S$. | From Intersection of Plane with Sphere is Circle it is confirmed that $C$ is indeed a circle.
{{Recall|Great Circle}}
{{:Definition:Great Circle}}
{{Recall|Small Circle}}
{{:Definition:Small Circle/Definition 1}}
The result follows directly.
{{qed}} | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]].
Let $P$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $S$ but is not [[Definition:Tangent Plane|tangent]] to $S$.
Let $C$ be the [[Definition:Circle|circle]] formed by the [[Definition:Intersection (Geometry)|intersection]... | From [[Intersection of Plane with Sphere is Circle]] it is confirmed that $C$ is indeed a [[Definition:Circle|circle]].
{{Recall|Great Circle}}
{{:Definition:Great Circle}}
{{Recall|Small Circle}}
{{:Definition:Small Circle/Definition 1}}
The result follows directly.
{{qed}} | Intersection of Plane with Sphere is Great Circle iff Passing through Center | https://proofwiki.org/wiki/Intersection_of_Plane_with_Sphere_is_Great_Circle_iff_Passing_through_Center | https://proofwiki.org/wiki/Intersection_of_Plane_with_Sphere_is_Great_Circle_iff_Passing_through_Center | [
"Small Circles",
"Radii of Circles"
] | [
"Definition:Sphere/Geometry",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Tangent Plane",
"Definition:Circle",
"Definition:Intersection (Geometry)",
"Definition:Great Circle",
"Definition:Sphere/Geometry/Center",
"Definition:Small Circle"
] | [
"Intersection of Plane with Sphere is Circle",
"Definition:Circle"
] |
proofwiki-22378 | Measure of Total Solid Angle around Point | The total solid angle around a point has a measure of $4 \pi$ steradians. | Let $Q$ denote this total solid angle
{{Recall|steradian}}
{{:Definition:Steradian}}
Hence $Q$ is equal to the total surface area of $S$ divided by $r^2$.
That is:
{{begin-eqn}}
{{eqn | l = Q
| r = \dfrac A {r^2}
| c = where $A$ is the total surface area of $S$
}}
{{eqn | r = \dfrac {4 \pi r^2} {r^2}
... | The total [[Definition:Solid Angle|solid angle]] around a [[Definition:Point|point]] has a [[Definition:Measurement|measure]] of $4 \pi$ [[Definition:Steradian|steradians]]. | Let $Q$ denote this total [[Definition:Solid Angle|solid angle]]
{{Recall|steradian}}
{{:Definition:Steradian}}
Hence $Q$ is equal to the total [[Definition:Surface Area|surface area]] of $S$ divided by $r^2$.
That is:
{{begin-eqn}}
{{eqn | l = Q
| r = \dfrac A {r^2}
| c = where $A$ is the total [[Defin... | Measure of Total Solid Angle around Point | https://proofwiki.org/wiki/Measure_of_Total_Solid_Angle_around_Point | https://proofwiki.org/wiki/Measure_of_Total_Solid_Angle_around_Point | [
"Solid Angles"
] | [
"Definition:Solid Angle",
"Definition:Point",
"Definition:Measurable Property/Measurement",
"Definition:Steradian"
] | [
"Definition:Solid Angle",
"Definition:Surface Area",
"Definition:Surface Area",
"Surface Area of Sphere"
] |
proofwiki-22379 | Measure of Trihedral Angle formed by 3 Mutually Perpendicular Half-Lines | Let $Q$ be the trihedral angle formed by $3$ half-lines each of which is perpendicular to the other two half-lines.
Then $Q$ measures $\dfrac \pi 2$ steradians. | {{ProofWanted|Prove that $Q$ subtends $1/8$ of a sphere then use Measure of Total Solid Angle around Point}} | Let $Q$ be the [[Definition:Trihedral Angle|trihedral angle]] formed by $3$ [[Definition:Half-Line|half-lines]] each of which is [[Definition:Perpendicular Lines|perpendicular]] to the other two [[Definition:Half-Line|half-lines]].
Then $Q$ measures $\dfrac \pi 2$ [[Definition:Steradian|steradians]]. | {{ProofWanted|Prove that $Q$ subtends $1/8$ of a sphere then use [[Measure of Total Solid Angle around Point]]}} | Measure of Trihedral Angle formed by 3 Mutually Perpendicular Half-Lines | https://proofwiki.org/wiki/Measure_of_Trihedral_Angle_formed_by_3_Mutually_Perpendicular_Half-Lines | https://proofwiki.org/wiki/Measure_of_Trihedral_Angle_formed_by_3_Mutually_Perpendicular_Half-Lines | [
"Trihedral Angles",
"Solid Angles"
] | [
"Definition:Trihedral Angle",
"Definition:Line/Infinite Half-Line",
"Definition:Right Angle/Perpendicular",
"Definition:Line/Infinite Half-Line",
"Definition:Steradian"
] | [
"Measure of Total Solid Angle around Point"
] |
proofwiki-22380 | Symmetry Group of Equilateral Triangle is Solvable | The symmetry group $D_3$ of the equilateral triangle is a solvable group. | From Symmetry Group of Equilateral Triangle is Symmetric Group, $D_3$ is the symmetric group on $3$ letters $S_3$.
The result follows from Examples of Composition Series: Symmetric Group $S_3$.
{{qed}} | The [[Definition:Symmetry Group of Equilateral Triangle|symmetry group]] $D_3$ of the [[Definition:Equilateral Triangle|equilateral triangle]] is a [[Definition:Solvable Group|solvable group]]. | From [[Symmetry Group of Equilateral Triangle is Symmetric Group]], $D_3$ is the [[Definition:Symmetric Group|symmetric group on $3$ letters]] $S_3$.
The result follows from [[Composition Series/Examples/Symmetric Group S3|Examples of Composition Series: Symmetric Group $S_3$]].
{{qed}} | Symmetry Group of Equilateral Triangle is Solvable | https://proofwiki.org/wiki/Symmetry_Group_of_Equilateral_Triangle_is_Solvable | https://proofwiki.org/wiki/Symmetry_Group_of_Equilateral_Triangle_is_Solvable | [
"Symmetry Group of Equilateral Triangle",
"Examples of Solvable Groups"
] | [
"Definition:Symmetry Group of Equilateral Triangle",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Solvable Group"
] | [
"Symmetry Group of Equilateral Triangle is Symmetric Group",
"Definition:Symmetric Group",
"Composition Series/Examples/Symmetric Group S3"
] |
proofwiki-22381 | Condition for Group to be Solvable | Let $G$ be a group.
Let $G$ be such that:
:$G$ has a normal subgroup $H$ which is solvable
:the quotient group $G / H$ is cyclic.
Then $G$ is a solvable group. | {{Recall|Solvable Group}}
{{:Definition:Solvable Group}}
Let $\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_n = H$ be a composition series in which each factor is a cyclic group.
By assumption, the above composition series could be extended to $\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_n \lhd G$ in which each factor is still... | Let $G$ be a [[Definition:Group|group]].
Let $G$ be such that:
:$G$ has a [[Definition:Normal Subgroup|normal subgroup]] $H$ which is [[Definition:Solvable Group|solvable]]
:the [[Definition:Quotient Group|quotient group]] $G / H$ is [[Definition:Cyclic Group|cyclic]].
Then $G$ is a [[Definition:Solvable Group|solvab... | {{Recall|Solvable Group}}
{{:Definition:Solvable Group}}
Let $\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_n = H$ be a [[Definition:Composition Series|composition series]] in which each [[Definition:Factor of Normal Series|factor]] is a [[Definition:Cyclic Group|cyclic group]].
By assumption, the above [[Definition:Compo... | Condition for Group to be Solvable | https://proofwiki.org/wiki/Condition_for_Group_to_be_Solvable | https://proofwiki.org/wiki/Condition_for_Group_to_be_Solvable | [
"Solvable Groups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Solvable Group",
"Definition:Quotient Group",
"Definition:Cyclic Group",
"Definition:Solvable Group"
] | [
"Definition:Composition Series",
"Definition:Normal Series/Factor Group",
"Definition:Cyclic Group",
"Definition:Composition Series",
"Definition:Normal Series/Factor Group",
"Definition:Cyclic Group"
] |
proofwiki-22382 | Stone Space of Boolean Lattice is Hausdorff | Let $B = \struct {S, \preceq}$ be a Boolean lattice.
Let $\map S B = \struct {U, \tau}$ be the Stone space of $B$.
Then, $\map S B$ is a Hausdorff space. | Let $x, y \in U$ be arbitrary, with $x \ne y$.
By definition of set equality, there is some $a \in S$ such that:
:$\neg \paren {a \in x \iff a \in y}$
{{WLOG}}, assume that $a \in x$ and $a \notin y$.
Now, let:
:$X = \set {z \in U : a \in z}$
:$Y = \set {z \in U : \paren {\neg a} \in z}$
By definition of the Stone spac... | Let $B = \struct {S, \preceq}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $\map S B = \struct {U, \tau}$ be the [[Definition:Stone Space|Stone space]] of $B$.
Then, $\map S B$ is a [[Definition:Hausdorff Space|Hausdorff space]]. | Let $x, y \in U$ be arbitrary, with $x \ne y$.
By definition of [[Definition:Set Equality|set equality]], there is some $a \in S$ such that:
:$\neg \paren {a \in x \iff a \in y}$
{{WLOG}}, assume that $a \in x$ and $a \notin y$.
Now, let:
:$X = \set {z \in U : a \in z}$
:$Y = \set {z \in U : \paren {\neg a} \in z}$
... | Stone Space of Boolean Lattice is Hausdorff | https://proofwiki.org/wiki/Stone_Space_of_Boolean_Lattice_is_Hausdorff | https://proofwiki.org/wiki/Stone_Space_of_Boolean_Lattice_is_Hausdorff | [
"Boolean Lattices",
"Hausdorff Spaces",
"Stone Spaces"
] | [
"Definition:Boolean Lattice",
"Definition:Stone Space",
"Definition:T2 Space"
] | [
"Definition:Set Equality",
"Definition:Stone Space",
"Definition:Open Set/Topology",
"Ultrafilter on Boolean Lattice Contains Element or Complement",
"Definition:Filter",
"Definition:Ultrafilter (Order Theory)",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:T2 Space",
"Ca... |
proofwiki-22383 | Ambiguous Case for Spherical Triangle/Side-Side-Angle | Let the sides $a$ and $b$ be known.
Let the angle $\sphericalangle B$ also be known.
Then it may not be possible to know the value of $\sphericalangle A$. | From the Spherical Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin A} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is n... | Let the [[Definition:Side of Spherical Triangle|sides]] $a$ and $b$ be known.
Let the [[Definition:Spherical Angle|angle]] $\sphericalangle B$ also be known.
Then it may not be possible to know the value of $\sphericalangle A$. | From the [[Spherical Law of Sines]], we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin A} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$... | Ambiguous Case for Spherical Triangle/Side-Side-Angle | https://proofwiki.org/wiki/Ambiguous_Case_for_Spherical_Triangle/Side-Side-Angle | https://proofwiki.org/wiki/Ambiguous_Case_for_Spherical_Triangle/Side-Side-Angle | [
"Ambiguous Case for Spherical Triangle"
] | [
"Definition:Spherical Triangle/Side",
"Definition:Spherical Angle"
] | [
"Spherical Law of Sines"
] |
proofwiki-22384 | Ambiguous Case for Spherical Triangle/Angle-Angle-Side | Let the angles $\sphericalangle A$ and $\sphericalangle B$ be known.
Let the side $b$ also be known.
Then it may not be possible to know the value of $a$. | From the Spherical Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin a = \dfrac {\sin b \sin A} {\sin B}$
We find that $0 < \sin a \le 1$.
We have that:
:$\sin a = \map \sin {\pi - a}$
and so unless $\sin a = 1$ and so $a = \dfrac \pi 2$, it is n... | Let the [[Definition:Spherical Angle|angles]] $\sphericalangle A$ and $\sphericalangle B$ be known.
Let the [[Definition:Side of Spherical Triangle|side]] $b$ also be known.
Then it may not be possible to know the value of $a$. | From the [[Spherical Law of Sines]], we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin a = \dfrac {\sin b \sin A} {\sin B}$
We find that $0 < \sin a \le 1$.
We have that:
:$\sin a = \map \sin {\pi - a}$
and so unless $\sin a = 1$ and so $a = \dfrac \pi 2$... | Ambiguous Case for Spherical Triangle/Angle-Angle-Side | https://proofwiki.org/wiki/Ambiguous_Case_for_Spherical_Triangle/Angle-Angle-Side | https://proofwiki.org/wiki/Ambiguous_Case_for_Spherical_Triangle/Angle-Angle-Side | [
"Ambiguous Case for Spherical Triangle"
] | [
"Definition:Spherical Angle",
"Definition:Spherical Triangle/Side"
] | [
"Spherical Law of Sines"
] |
proofwiki-22385 | Group Automorphism is Endomorphism | Let $G$ a group.
Let $\phi: G \to G$ be a (group) automorphism on $G$.
Then $\phi$ is a group endomorphism. | {{Recall|Group Automorphism}}
{{:Definition:Group Automorphism}}
{{Recall|Group Isomorphism}}
{{:Definition:Group Isomorphism}}
{{Recall|Group Endomorphism}}
{{:Definition:Group Endomorphism}}
We have been given that $\phi$ is a group automorphism.
Hence {{afortiori}} $\phi$ is a group homomorphism from $G$ to $G$.
Hen... | Let $G$ a [[Definition:Group|group]].
Let $\phi: G \to G$ be a [[Definition:Group Automorphism|(group) automorphism]] on $G$.
Then $\phi$ is a [[Definition:Group Endomorphism|group endomorphism]]. | {{Recall|Group Automorphism}}
{{:Definition:Group Automorphism}}
{{Recall|Group Isomorphism}}
{{:Definition:Group Isomorphism}}
{{Recall|Group Endomorphism}}
{{:Definition:Group Endomorphism}}
We have been given that $\phi$ is a [[Definition:Group Automorphism|group automorphism]].
Hence {{afortiori}} $\phi$ is a ... | Group Automorphism is Endomorphism/Proof 1 | https://proofwiki.org/wiki/Group_Automorphism_is_Endomorphism | https://proofwiki.org/wiki/Group_Automorphism_is_Endomorphism/Proof_1 | [
"Group Automorphism is Endomorphism",
"Group Automorphisms",
"Group Endomorphisms"
] | [
"Definition:Group",
"Definition:Group Automorphism",
"Definition:Group Endomorphism"
] | [
"Definition:Group Automorphism",
"Definition:Group Homomorphism",
"Definition:Group Endomorphism"
] |
proofwiki-22386 | Fully Characteristic Subgroup is Characteristic | Let $G$ a group.
Let $H$ a fully characteristic subgroup of $G$.
Then $H$ is a characteristic subgroup of $G$. | By definition of a fully characteristic subgroup,
$H$ is invariant under every group endomorphism of $G$.
By Group Automorphism is Endomorphism, every group automorphism is a group endomorphism.
Hence, invariance over every group endomorphism implies invariance over every group automorphism.
And by definition of a char... | Let $G$ a [[Definition:Group|group]].
Let $H$ a [[Definition:Fully Characteristic Subgroup|fully characteristic subgroup]] of $G$.
Then $H$ is a [[Definition:Characteristic Subgroup|characteristic subgroup]] of $G$. | By [[Definition:Fully Characteristic Subgroup|definition of a fully characteristic subgroup]],
$H$ is invariant under every [[Definition:Group Endomorphism|group endomorphism]] of $G$.
By [[Group Automorphism is Endomorphism]], every [[Definition:Group Automorphism|group automorphism]] is a [[Definition:Group Endomorp... | Fully Characteristic Subgroup is Characteristic/Proof 1 | https://proofwiki.org/wiki/Fully_Characteristic_Subgroup_is_Characteristic | https://proofwiki.org/wiki/Fully_Characteristic_Subgroup_is_Characteristic/Proof_1 | [
"Fully Characteristic Subgroup is Characteristic",
"Characteristic Subgroups"
] | [
"Definition:Group",
"Definition:Fully Characteristic Subgroup",
"Definition:Characteristic Subgroup"
] | [
"Definition:Fully Characteristic Subgroup",
"Definition:Group Endomorphism",
"Group Automorphism is Endomorphism",
"Definition:Group Automorphism",
"Definition:Group Endomorphism",
"Definition:Group Endomorphism",
"Definition:Group Automorphism",
"Definition:Characteristic Subgroup"
] |
proofwiki-22387 | Proper Ideal is Prime iff Contains Element or Complement | Let $\struct {S, \preceq}$ be a boolean lattice.
Let $I$ be a proper ideal on $S$.
Then:
:$I$ is a prime ideal
{{iff}}:
:for every $a \in S$, either $a \in I$ or $\neg a \in I$. | === Necessary Condition ===
Suppose that $I$ is a prime ideal.
Then, by definition:
:$S \setminus I$ is a filter
{{AimForCont}} $a, \neg a \notin I$.
Then, $a, \neg \in S \setminus I$.
Hence, by Filter is Closed under Meet:
:$\neg a \wedge a \in S \setminus I$
but then by definition of complement:
:$\bot = \neg \wedge ... | Let $\struct {S, \preceq}$ be a [[Definition:Boolean Lattice|boolean lattice]].
Let $I$ be a [[Definition:Proper Ideal (Order Theory)|proper ideal]] on $S$.
Then:
:$I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]]
{{iff}}:
:for every $a \in S$, either $a \in I$ or $\neg a \in I$. | === Necessary Condition ===
Suppose that $I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]].
Then, by definition:
:$S \setminus I$ is a [[Definition:Filter|filter]]
{{AimForCont}} $a, \neg a \notin I$.
Then, $a, \neg \in S \setminus I$.
Hence, by [[Filter is Closed under Meet]]:
:$\neg a \wedge a \in S... | Proper Ideal is Prime iff Contains Element or Complement | https://proofwiki.org/wiki/Proper_Ideal_is_Prime_iff_Contains_Element_or_Complement | https://proofwiki.org/wiki/Proper_Ideal_is_Prime_iff_Contains_Element_or_Complement | [
"Boolean Lattices",
"Prime Ideals (Order Theory)"
] | [
"Definition:Boolean Lattice",
"Definition:Ideal (Order Theory)/Proper Ideal",
"Definition:Prime Ideal (Order Theory)"
] | [
"Definition:Prime Ideal (Order Theory)",
"Definition:Filter",
"Filter is Closed under Meet",
"Definition:Complement (Lattice Theory)",
"Bottom in Ideal"
] |
proofwiki-22388 | Group whose Derived Subgroup is Trivial is Abelian | Let $G$ be a group with identity $e$.
Let $G'$ be the derived subgroup of $G$.
Let $G'$ be the trivial group $\set e$.
Then $G$ is abelian. | We assume {{hypothesis}} that the derived subgroup of $G$ is $\set e$.
{{AimForCont}} $G$ is non-abelian.
Hence by definition:
:$\exists g, h \in G: g h \ne h g$
Multiplying both sides by $g^{-1} h^{-1}$:
:$\exists g, h \in G: g^{-1} h^{-1} g h \ne e$
But from the definition of the derived subgroup:
:$G' := \set {g^{-1... | Let $G$ be a [[Definition:Group|group]] with identity $e$.
Let $G'$ be the [[Definition:Derived Subgroup|derived subgroup]] of $G$.
Let $G'$ be the [[Definition:Trivial Group|trivial group]] $\set e$.
Then $G$ is [[Definition:Abelian Group|abelian]]. | We assume {{hypothesis}} that the [[Definition:Derived Subgroup|derived subgroup]] of $G$ is $\set e$.
{{AimForCont}} $G$ is non-[[Definition:Abelian Group|abelian]].
Hence by definition:
:$\exists g, h \in G: g h \ne h g$
Multiplying both sides by $g^{-1} h^{-1}$:
:$\exists g, h \in G: g^{-1} h^{-1} g h \ne e$
But... | Group whose Derived Subgroup is Trivial is Abelian | https://proofwiki.org/wiki/Group_whose_Derived_Subgroup_is_Trivial_is_Abelian | https://proofwiki.org/wiki/Group_whose_Derived_Subgroup_is_Trivial_is_Abelian | [
"Derived Subgroups",
"Abelian Groups",
"Trivial Group"
] | [
"Definition:Group",
"Definition:Derived Subgroup",
"Definition:Trivial Group",
"Definition:Abelian Group"
] | [
"Definition:Derived Subgroup",
"Definition:Abelian Group",
"Definition:Derived Subgroup",
"Definition:Trivial Group",
"Proof by Contraposition",
"Definition:Abelian Group"
] |
proofwiki-22389 | Boolean Prime Ideal Theorem/Element Extension Lemma | Let $\struct {B, \vee, \wedge, \le}$ be a Boolean lattice.
Let $F \subseteq B$ be a filter on $B$.
Let $a, x \in B$ such that:
:$a \notin F$
Then either:
:$a \vee x \notin F$
or:
:$a \vee \neg x \notin F$ | {{AimForCont}} that both:
:$a \vee x \in F$
and:
:$a \vee \neg x \in F$
Then:
{{begin-eqn}}
{{eqn | l = F
| o = \ni
| r = \paren {a \vee x} \wedge \paren {a \vee \neg x}
| c = Filter is Closed under Meet
}}
{{eqn | r = \paren {a \wedge \paren {a \vee \neg x} } \vee \paren {x \wedge \paren {a \vee \neg... | Let $\struct {B, \vee, \wedge, \le}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $F \subseteq B$ be a [[Definition:Filter|filter]] on $B$.
Let $a, x \in B$ such that:
:$a \notin F$
Then either:
:$a \vee x \notin F$
or:
:$a \vee \neg x \notin F$ | {{AimForCont}} that both:
:$a \vee x \in F$
and:
:$a \vee \neg x \in F$
Then:
{{begin-eqn}}
{{eqn | l = F
| o = \ni
| r = \paren {a \vee x} \wedge \paren {a \vee \neg x}
| c = [[Filter is Closed under Meet]]
}}
{{eqn | r = \paren {a \wedge \paren {a \vee \neg x} } \vee \paren {x \wedge \paren {a \vee... | Boolean Prime Ideal Theorem/Element Extension Lemma | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem/Element_Extension_Lemma | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem/Element_Extension_Lemma | [
"Boolean Lattices"
] | [
"Definition:Boolean Lattice",
"Definition:Filter"
] | [
"Filter is Closed under Meet",
"Axiom:Distributive Lattice Axioms",
"Meet Absorbs Join",
"Axiom:Distributive Lattice Axioms",
"Meet is Commutative",
"Join Absorbs Meet",
"Definition:Contradiction",
"Category:Boolean Lattices"
] |
proofwiki-22390 | Interior of Complement equals Complement of Closure | Let $T$ be a topological space.
Let $H \subseteq T$.
Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$.
Let $\map \complement H$ be the complement of $H$ in $T$:
:$\map \complement H = T \setminus H$
Then:
:$\paren {\map \complement H}^\circ = \map \complement {H^-}$
This can alternatively be... | {{begin-eqn}}
{{eqn | l = H^\circ
| r = T \setminus \paren {T \setminus H^\circ}
| c = Relative Complement of Relative Complement
}}
{{eqn | r = T \setminus \paren {\paren {T \setminus H}^-}
| c = Complement of Interior equals Closure of Complement
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn | l = \par... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq T$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$ and $H^\circ$ denote the [[Definition:Interior (Topology)|interior]] of $H$.
Let $\map \complement H$ be the [[Definition:Relative Complement|complement of $H$ i... | {{begin-eqn}}
{{eqn | l = H^\circ
| r = T \setminus \paren {T \setminus H^\circ}
| c = [[Relative Complement of Relative Complement]]
}}
{{eqn | r = T \setminus \paren {\paren {T \setminus H}^-}
| c = [[Complement of Interior equals Closure of Complement]]
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn ... | Interior of Complement equals Complement of Closure | https://proofwiki.org/wiki/Interior_of_Complement_equals_Complement_of_Closure | https://proofwiki.org/wiki/Interior_of_Complement_equals_Complement_of_Closure | [
"Set Closures",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Relative Complement"
] | [
"Relative Complement of Relative Complement",
"Complement of Interior equals Closure of Complement",
"Complement of Interior equals Closure of Complement",
"Relative Complement of Relative Complement"
] |
proofwiki-22391 | Vector Subspace contains Zero Vector | Let $V$ be a vector space over a division ring $K$ whose unity is $1_K$.
Let $U \subseteq V$ be a subspace of $V$.
Then $U$ contains the zero vector of $V$. | Demonstrated during the course of the One-Step Vector Subspace Test.
{{qed}} | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Division Ring|division ring]] $K$ whose [[Definition:Unity of Ring|unity]] is $1_K$.
Let $U \subseteq V$ be a [[Definition:Vector Subspace|subspace]] of $V$.
Then $U$ contains the [[Definition:Zero Vector|zero vector]] of $V$. | Demonstrated during the course of the [[One-Step Vector Subspace Test]].
{{qed}} | Vector Subspace contains Zero Vector | https://proofwiki.org/wiki/Vector_Subspace_contains_Zero_Vector | https://proofwiki.org/wiki/Vector_Subspace_contains_Zero_Vector | [
"Vector Subspaces"
] | [
"Definition:Vector Space",
"Definition:Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Vector Subspace",
"Definition:Zero Vector"
] | [
"One-Step Vector Subspace Test"
] |
proofwiki-22392 | Singleton of Greatest Element in Two is Completely Prime Filter | Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the (Boolean Lattice) $\mathbf 2$.
Then:
:$\set \top$ is a completely prime filter | By definition of (Boolean Lattice) $\mathbf 2$:
:$\mathbf 2 := \set {\bot, \top}$
endowed with the logical operations $\lor$, $\land$ defined by the following Cayley tables:
:<nowiki>$\begin{array}{c|cc}
\lor & \bot & \top \\ \hline
\bot & \bot & \top \\
\top & \top & \top
\end{array} \qquad \begin{array}{c|cc}
\la... | Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]].
Then:
:$\set \top$ is a [[Definition:Completely Prime Filter|completely prime filter]] | By definition of [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]]:
:$\mathbf 2 := \set {\bot, \top}$
endowed with the logical operations $\lor$, $\land$ defined by the following [[Definition:Cayley Table|Cayley tables]]:
:<nowiki>$\begin{array}{c|cc}
\lor & \bot & \top \\ \hline
\bot & \bot & \top \... | Singleton of Greatest Element in Two is Completely Prime Filter | https://proofwiki.org/wiki/Singleton_of_Greatest_Element_in_Two_is_Completely_Prime_Filter | https://proofwiki.org/wiki/Singleton_of_Greatest_Element_in_Two_is_Completely_Prime_Filter | [
"Completely Prime Filters"
] | [
"Definition:Two (Boolean Lattice)",
"Definition:Completely Prime Filter"
] | [
"Definition:Two (Boolean Lattice)",
"Definition:Cayley Table",
"Definition:Ordering",
"Two is a Locale",
"Definition:Locale (Lattice Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Complete Lattice",
"Characterization of Completely Prime Filter in Complete Lattice",
"Definition:Finite",
... |
proofwiki-22393 | Supplemental Chords are Perpendicular | Let $\CC$ be a circle.
Let $K_1$ and $K_2$ be a pair of supplemental chords of $\CC$.
Then $K_1$ and $K_2$ are perpendicular. | This is merely a restatement of Thales' Theorem stated in terms of supplemental chords.
{{qed}} | Let $\CC$ be a [[Definition:Circle|circle]].
Let $K_1$ and $K_2$ be a [[Definition:Doubleton|pair]] of [[Definition:Supplemental Chords|supplemental chords]] of $\CC$.
Then $K_1$ and $K_2$ are [[Definition:Perpendicular Lines|perpendicular]]. | This is merely a restatement of [[Thales' Theorem]] stated in terms of [[Definition:Supplemental Chords|supplemental chords]].
{{qed}} | Supplemental Chords are Perpendicular | https://proofwiki.org/wiki/Supplemental_Chords_are_Perpendicular | https://proofwiki.org/wiki/Supplemental_Chords_are_Perpendicular | [
"Supplemental Chords"
] | [
"Definition:Circle",
"Definition:Doubleton",
"Definition:Supplemental Chords",
"Definition:Right Angle/Perpendicular"
] | [
"Thales' Theorem",
"Definition:Supplemental Chords"
] |
proofwiki-22394 | Squares which are Sum of Two Cubes/Formulation 1 | {{begin-eqn}}
{{eqn | l = A
| r = 3 n^3 + 6 n^2 - n
}}
{{eqn | l = B
| r = -3 n^3 + 6 n^2 + n
}}
{{eqn | l = C
| r = 6 n^2 \paren {3 n^2 + 1}
}}
{{end-eqn}}
where $n \in \C$. | Assume that:
:$A + B = 12 n^2$
Thus, factoring $A^3 + B^3$, we find that:
{{begin-eqn}}
{{eqn | l = A^3 + B^3
| r = \paren {A + B} \paren {A^2 - A B + B^2}
| c = Sum of Two Cubes
}}
{{eqn | ll = \leadsto
| l = \frac {\paren {A^3 + B^3} } {\paren {A + B} }
| r = \paren {A^2 - A B + B^2}
| ... | {{begin-eqn}}
{{eqn | l = A
| r = 3 n^3 + 6 n^2 - n
}}
{{eqn | l = B
| r = -3 n^3 + 6 n^2 + n
}}
{{eqn | l = C
| r = 6 n^2 \paren {3 n^2 + 1}
}}
{{end-eqn}}
where $n \in \C$. | Assume that:
:$A + B = 12 n^2$
Thus, factoring $A^3 + B^3$, we find that:
{{begin-eqn}}
{{eqn | l = A^3 + B^3
| r = \paren {A + B} \paren {A^2 - A B + B^2}
| c = [[Sum of Two Cubes]]
}}
{{eqn | ll = \leadsto
| l = \frac {\paren {A^3 + B^3} } {\paren {A + B} }
| r = \paren {A^2 - A B + B^2}
... | Squares which are Sum of Two Cubes/Formulation 1 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_1 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_1 | [
"Sums of Cubes"
] | [] | [
"Sum of Two Odd Powers/Examples/Sum of Two Cubes",
"Square of Difference",
"Square of Sum"
] |
proofwiki-22395 | Squares which are Sum of Two Cubes/Formulation 2 | {{begin-eqn}}
{{eqn | l = A
| r = \dfrac 1 4 \paren {3 n^3 + 6 n^2 - n}
}}
{{eqn | l = B
| r = \dfrac 1 4 \paren {-3 n^3 + 6 n^2 + n}
}}
{{eqn | l = C
| r = \dfrac 1 8 \paren {6 n^2 \paren {3 n^2 + 1} }
}}
{{end-eqn}}
where $n \in \C$. | {{begin-eqn}}
{{eqn | l = \paren {\frac A 4}^3 + \paren {\frac B 4}^3
| r = \frac 1 {64} \paren {A^3 + B^3}
| c =
}}
{{eqn | r = \paren {\frac C 8}^2
| c =
}}
{{end-eqn}}
From Squares which are Sum of Two Cubes: Formulation 1, we have:
{{:Squares which are Sum of Two Cubes/Formulation 1}}
Therefore:
{{... | {{begin-eqn}}
{{eqn | l = A
| r = \dfrac 1 4 \paren {3 n^3 + 6 n^2 - n}
}}
{{eqn | l = B
| r = \dfrac 1 4 \paren {-3 n^3 + 6 n^2 + n}
}}
{{eqn | l = C
| r = \dfrac 1 8 \paren {6 n^2 \paren {3 n^2 + 1} }
}}
{{end-eqn}}
where $n \in \C$. | {{begin-eqn}}
{{eqn | l = \paren {\frac A 4}^3 + \paren {\frac B 4}^3
| r = \frac 1 {64} \paren {A^3 + B^3}
| c =
}}
{{eqn | r = \paren {\frac C 8}^2
| c =
}}
{{end-eqn}}
From [[Squares which are Sum of Two Cubes/Formulation 1|Squares which are Sum of Two Cubes: Formulation 1]], we have:
{{:Squares w... | Squares which are Sum of Two Cubes/Formulation 2 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_2 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_2 | [
"Sums of Cubes"
] | [] | [
"Squares which are Sum of Two Cubes/Formulation 1",
"Category:Sums of Cubes"
] |
proofwiki-22396 | Equivalence of Definitions of Symmetric Matrices | {{TFAE|def = Symmetric Matrix}}
Let $\mathbf A$ be a square matrix over a set $S$. | Place the elements of $\mathbf A$ onto the $x y$ plane such that element $a_{i j}$ is at coordinate $\tuple {j, -i}$.
The indices are reversed because the vertical coordinate--the row index--is listed first in matrix element notation.
The negative sign on the $i$ index is because the rows of a matrix are counted from t... | {{TFAE|def = Symmetric Matrix}}
Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over a [[Definition:Set|set]] $S$. | Place the [[Definition:Element of Matrix|elements]] of $\mathbf A$ onto the [[Definition:XY Plane|$x y$ plane]] such that [[Definition:Element of Matrix|element]] $a_{i j}$ is at [[Definition:Cartesian Coordinate System|coordinate]] $\tuple {j, -i}$.
The [[Definition:Index of Matrix Element|indices]] are reversed beca... | Equivalence of Definitions of Symmetric Matrices | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Matrices | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Matrices | [
"Symmetric Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Set"
] | [
"Definition:Matrix/Element",
"Definition:Cartesian Plane",
"Definition:Matrix/Element",
"Definition:Cartesian Coordinate System",
"Definition:Matrix/Indices",
"Definition:Cartesian Coordinate System/Y Coordinate",
"Definition:Matrix/Row",
"Definition:Matrix/Indices",
"Definition:Matrix/Indices",
"... |
proofwiki-22397 | Squares which are Sum of Two Cubes/Formulation 3 | {{begin-eqn}}
{{eqn | l = A
| r = k^2 \paren {3 n^3 + 6 n^2 - n}
}}
{{eqn | l = B
| r = k^2 \paren {-3 n^3 + 6 n^2 + n}
}}
{{eqn | l = C
| r = k^3 \paren {6 n^2 \paren {3 n^2 + 1} }
}}
{{end-eqn}}
where $n, k \in \C$. | {{begin-eqn}}
{{eqn | l = \paren {k^2 A}^3 + \paren {k^2 B}^3
| r = k^6 \paren {A^3 + B^3}
| c =
}}
{{eqn | r = \paren {k^3 C}^2
| c =
}}
{{end-eqn}}
From Squares which are Sum of Two Cubes: Formulation 1, we have:
{{:Squares which are Sum of Two Cubes/Formulation 1}}
Therefore:
{{begin-eqn}}
{{eqn | l... | {{begin-eqn}}
{{eqn | l = A
| r = k^2 \paren {3 n^3 + 6 n^2 - n}
}}
{{eqn | l = B
| r = k^2 \paren {-3 n^3 + 6 n^2 + n}
}}
{{eqn | l = C
| r = k^3 \paren {6 n^2 \paren {3 n^2 + 1} }
}}
{{end-eqn}}
where $n, k \in \C$. | {{begin-eqn}}
{{eqn | l = \paren {k^2 A}^3 + \paren {k^2 B}^3
| r = k^6 \paren {A^3 + B^3}
| c =
}}
{{eqn | r = \paren {k^3 C}^2
| c =
}}
{{end-eqn}}
From [[Squares which are Sum of Two Cubes/Formulation 1|Squares which are Sum of Two Cubes: Formulation 1]], we have:
{{:Squares which are Sum of Two C... | Squares which are Sum of Two Cubes/Formulation 3 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_3 | https://proofwiki.org/wiki/Squares_which_are_Sum_of_Two_Cubes/Formulation_3 | [
"Sums of Cubes"
] | [] | [
"Squares which are Sum of Two Cubes/Formulation 1",
"Category:Sums of Cubes"
] |
proofwiki-22398 | Inclusion Functor is Functor | Let $\mathbf D$ be a metacategory, and let $\mathbf C$ be a subcategory of $\mathbf D$.
Let $\Iota_{\mathbf C}: \mathbf C \to \mathbf D$ be the inclusion functor on $\mathbf C$
Then:
:$\Iota_{\mathbf C}$ is a (covariant) functor | Let $f, g$ be morphisms of $\mathbf C$ such that $g \circ f$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \map {\Iota_{\mathbf C} } {g \circ f}
| r = g \circ f
| c = {{Defof|Inclusion Functor}}
}}
{{eqn | r = \map {\Iota_{\mathbf C} } g \circ \map {\Iota_{\mathbf C} } f
| c = {{Defof|Inclusion Functor}... | Let $\mathbf D$ be a [[Definition:Metacategory|metacategory]], and let $\mathbf C$ be a [[Definition:Subcategory|subcategory]] of $\mathbf D$.
Let $\Iota_{\mathbf C}: \mathbf C \to \mathbf D$ be the [[Definition:Inclusion Functor|inclusion functor on $\mathbf C$]]
Then:
:$\Iota_{\mathbf C}$ is a [[Definition:Covaria... | Let $f, g$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C$ such that $g \circ f$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \map {\Iota_{\mathbf C} } {g \circ f}
| r = g \circ f
| c = {{Defof|Inclusion Functor}}
}}
{{eqn | r = \map {\Iota_{\mathbf C} } g \circ \map {\Iota_{\mathbf C... | Inclusion Functor is Functor | https://proofwiki.org/wiki/Inclusion_Functor_is_Functor | https://proofwiki.org/wiki/Inclusion_Functor_is_Functor | [
"Functors"
] | [
"Definition:Metacategory",
"Definition:Subcategory",
"Definition:Inclusion Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Morphism",
"Definition:Object",
"Definition:Functor/Covariant"
] |
proofwiki-22399 | Recurrence Formula for Chebyshev Polynomials of the First Kind | Let $\map {T_n} x$ denote the Chebyshev polynomials of the first kind of order $n$.
Then:
:$\map {T_n} x = \begin {cases} 1 & : n = 0 \\ x & : n = 1 \\ 2 x \, \map {T_{n - 1} } x - \map {T_{n - 2} } x & : n > 1 \end {cases}$ | From the {{Defof|Chebyshev Polynomials/First Kind|Chebyshev Polynomials of the First Kind}}, we have:
:$\map {T_n} x = \map \cos {n \arccos x}$
For $n = 0$, we have:
{{begin-eqn}}
{{eqn | l = \map {T_0} x
| r = \map \cos {0 \arccos x}
| c = {{Defof|Chebyshev Polynomials/First Kind|Chebyshev Polynomials of t... | Let $\map {T_n} x$ denote the [[Definition:Chebyshev Polynomial of the First Kind|Chebyshev polynomials of the first kind of order $n$]].
Then:
:$\map {T_n} x = \begin {cases} 1 & : n = 0 \\ x & : n = 1 \\ 2 x \, \map {T_{n - 1} } x - \map {T_{n - 2} } x & : n > 1 \end {cases}$ | From the {{Defof|Chebyshev Polynomials/First Kind|Chebyshev Polynomials of the First Kind}}, we have:
:$\map {T_n} x = \map \cos {n \arccos x}$
For $n = 0$, we have:
{{begin-eqn}}
{{eqn | l = \map {T_0} x
| r = \map \cos {0 \arccos x}
| c = {{Defof|Chebyshev Polynomials/First Kind|Chebyshev Polynomials ... | Recurrence Formula for Chebyshev Polynomials of the First Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Chebyshev_Polynomials_of_the_First_Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Chebyshev_Polynomials_of_the_First_Kind | [
"Recurrence Formula for Chebyshev Polynomials of the First Kind",
"Chebyshev Polynomials of the First Kind"
] | [
"Definition:Chebyshev Polynomials/First Kind"
] | [
"Cosine of Zero is One"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.