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-18600 | Equivalence of Definitions of P-adic Valuation on P-adic Numbers | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
{{TFAE|def = P-adic Valuation on P-adic Numbers|view = the $p$-adic valuation on $\struct {\Q_p, \norm {\,\cdot\,}_p}$}} | Let $x \in \Q_p \setminus \set 0$.
Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion:
:$l = \min \set {i: i \ge m \land d_i \ne 0}$
From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
:$\norm x_p = p^{-l}$
By definition of real general logarithm:
:$-\log_p \no... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
{{TFAE|def = P-adic Valuation on P-adic Numbers|view = the $p$-adic valuation on $\struct {\Q_p, \norm {\,\cdot\,}_p}$}} | Let $x \in \Q_p \setminus \set 0$.
Let $l$ be the [[Definition:Index Variable of Summation|index]] of the first [[Definition:Zero (Number)|non-zero]] [[Definition:Coefficient of Power Series|coefficient]] in the [[Definition:P-adic Expansion|$p$-adic expansion]]:
:$l = \min \set {i: i \ge m \land d_i \ne 0}$
From [[P... | Equivalence of Definitions of P-adic Valuation on P-adic Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Valuation_on_P-adic_Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Valuation_on_P-adic_Numbers | [
"P-adic Valuation on P-adic Numbers"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number"
] | [
"Definition:Summation/Index Variable",
"Definition:Zero (Number)",
"Definition:Power Series/Coefficient",
"Definition:P-adic Expansion",
"P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient",
"Definition:General Logarithm/Positive Real",
"Category:P-adic Valuation on P-adic Number... |
proofwiki-18601 | Similarity Mapping is Automorphism | Let $G$ be a vector space over a field $\struct {K, +, \times}$.
Let $\beta \in K$.
Let $s_\beta: G \to G$ be the similarity on $G$ defined as:
:$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$
If $\beta \ne 0$ then $s_\beta$ is an automorphism of $G$. | By definition, a vector space automorphism on $G$ is a vector space isomorphism from $G$ to $G$ itself.
To prove that $s_\beta$ is a '''automorphism''' it is sufficient to demonstrate that:
By definition, a vector space isomorphism is a mapping $s_\beta: G \to G$ such that:
:$(1): \quad s_\beta$ is a bijection
:$(2): \... | Let $G$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $\struct {K, +, \times}$.
Let $\beta \in K$.
Let $s_\beta: G \to G$ be the [[Definition:Similarity Mapping|similarity]] on $G$ defined as:
:$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x... | By definition, a [[Definition:Vector Space Automorphism|vector space automorphism]] on $G$ is a [[Definition:Vector Space Isomorphism|vector space isomorphism]] from $G$ to $G$ itself.
To prove that $s_\beta$ is a '''[[Definition:Vector Space Automorphism|automorphism]]''' it is sufficient to demonstrate that:
By def... | Similarity Mapping is Automorphism | https://proofwiki.org/wiki/Similarity_Mapping_is_Automorphism | https://proofwiki.org/wiki/Similarity_Mapping_is_Automorphism | [
"Similarity Mappings",
"Automorphisms (Abstract Algebra)"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Similarity Mapping",
"Definition:Vector Space Automorphism"
] | [
"Definition:Vector Space Automorphism",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Definition:Vector Space Automorphism",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Definition:Mapping"... |
proofwiki-18602 | Inverse of Similarity Mapping | Let $G$ be a vector space over a field $K$.
Let $\beta \in K$ such that $\beta \ne 0$.
Let $s_\beta: G \to G$ be the similarity on $G$ defined as:
:$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$
Let $\paren {s_\beta}^{-1}$ denote the inverse of $s_\beta$.
Then:
:$\paren {s_\beta}^{-1} = s_{\bet... | From Similarity Mapping is Automorphism, $s_\beta$ is an automorphism of $G$.
Hence $s_\beta$ is an vector space isomorphism from $G$ to $G$ itself.
So by definition $s_\beta$ is a bijection.
Hence the existence of this inverse $\paren {s_\beta}^{-1}$ follows from Bijection iff Left and Right Inverse.
By {{Field-axiom|... | Let $G$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $\beta \in K$ such that $\beta \ne 0$.
Let $s_\beta: G \to G$ be the [[Definition:Similarity Mapping|similarity]] on $G$ defined as:
:$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathb... | From [[Similarity Mapping is Automorphism]], $s_\beta$ is an [[Definition:Vector Space Automorphism|automorphism]] of $G$.
Hence $s_\beta$ is an [[Definition:Vector Space Isomorphism|vector space isomorphism]] from $G$ to $G$ itself.
So by definition $s_\beta$ is a [[Definition:Bijection|bijection]].
Hence the exist... | Inverse of Similarity Mapping | https://proofwiki.org/wiki/Inverse_of_Similarity_Mapping | https://proofwiki.org/wiki/Inverse_of_Similarity_Mapping | [
"Similarity Mappings"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Similarity Mapping",
"Definition:Inverse Mapping",
"Definition:Multiplicative Inverse"
] | [
"Similarity Mapping is Automorphism",
"Definition:Vector Space Automorphism",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Bijection iff Left and Right Inverse",
"Definition:Multiplicative I... |
proofwiki-18603 | Existence of Orthonormal Frames | Let $\struct{M, g}$ be a $n$-dimensional Riemannian manifold with or without boundary.
Let $p \in M$ be a point.
Let $TM$ be the tangent bundle of $M$.
Let $U \subseteq M$ be an open subset.
Suppose $\tuple {X_j}$ is a smooth local frame for $TM$ over $U$.
Then for all $p \in M$ there is a smooth orthonormal frame $\t... | {{ProofWanted|Use Gram-Schmidt on vectors $X_j$; obtain orthonormal vector fields $E_j$; denominators are nonvanishing so $E_j$ are smooth; apply this to any smooth local frame}} | Let $\struct{M, g}$ be a [[Definition:Riemannian Manifold/Dimension|$n$-dimensional]] [[Definition:Riemannian Manifold|Riemannian manifold]] with or without [[Definition:Boundary (Topology)|boundary]].
Let $p \in M$ be a [[Definition:Point|point]].
Let $TM$ be the [[Definition:Tangent Bundle|tangent bundle]] of $M$.... | {{ProofWanted|Use Gram-Schmidt on vectors $X_j$; obtain orthonormal vector fields $E_j$; denominators are nonvanishing so $E_j$ are smooth; apply this to any smooth local frame}} | Existence of Orthonormal Frames | https://proofwiki.org/wiki/Existence_of_Orthonormal_Frames | https://proofwiki.org/wiki/Existence_of_Orthonormal_Frames | [
"Orthonormal Frames"
] | [
"Definition:Riemannian Manifold/Dimension",
"Definition:Riemannian Manifold",
"Definition:Boundary (Topology)",
"Definition:Point",
"Definition:Tangent Bundle",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Smooth Frame",
"Definition:Local Frame",
"Definition:Smooth Frame",
"D... | [] |
proofwiki-18604 | Value of Vandermonde Determinant/Formulation 1 | Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin{vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
1 & x_3 & {x_3}^2 & \cdots & {x_3}^{n - 2} & {x_3}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_{n - 1} & {x_{n - 1} }^2 & \... | Let $V_n$ be the '''[[Definition:Vandermonde Determinant/Formulation 1|Vandermonde determinant of order $n$]]''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin{vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
1 & x_3 & {x_3}^2 & \cdots & {x_3}^{n - 2} & {x_3}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_{n - 1} & {x_{n - 1} }^2 & \... | Value of Vandermonde Determinant/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1/Proof_1 | [
"Value of Vandermonde Determinant"
] | [
"Definition:Vandermonde Determinant/Formulation 1"
] | [
"Multiple of Row Added to Row of Determinant",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Determinant with Row Multiplied by Constant",
"Determinan... |
proofwiki-18605 | Value of Vandermonde Determinant/Formulation 1 | Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Proof by induction:
Let the Vandermonde determinant be presented in the following form:
:<nowiki>$V_n = \begin {vmatrix}
{x_1}^{n - 1} & {x_1}^{n - 2} & \cdots & x_1 & 1 \\
{x_2}^{n - 1} & {x_2}^{n - 2} & \cdots & x_2 & 1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
{x_n}^{n - 1} & {x_... | Let $V_n$ be the '''[[Definition:Vandermonde Determinant/Formulation 1|Vandermonde determinant of order $n$]]''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Proof by [[Principle of Mathematical Induction|induction]]:
Let the [[Definition:Vandermonde Determinant|Vandermonde determinant]] be presented in [[Definition:Vandermonde Determinant/Formulation 1/Also presented as/Ones at Right|the following form]]:
:<nowiki>$V_n = \begin {vmatrix}
{x_1}^{n - 1} & {x_1}^{n - 2} & \... | Value of Vandermonde Determinant/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1/Proof_2 | [
"Value of Vandermonde Determinant"
] | [
"Definition:Vandermonde Determinant/Formulation 1"
] | [
"Principle of Mathematical Induction",
"Definition:Vandermonde Determinant",
"Definition:Vandermonde Determinant/Formulation 1/Also presented as/Ones at Right",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Laplace Expa... |
proofwiki-18606 | Value of Vandermonde Determinant/Formulation 1 | Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin {vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
1 & x_3 & {x_3}^2 & \cdots & {x_3}^{n - 2} & {x_3}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_{n - 1} & {x_{n - 1} }^2 & ... | Let $V_n$ be the '''[[Definition:Vandermonde Determinant/Formulation 1|Vandermonde determinant of order $n$]]''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin {vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
1 & x_3 & {x_3}^2 & \cdots & {x_3}^{n - 2} & {x_3}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_{n - 1} & {x_{n - 1} }^2 & ... | Value of Vandermonde Determinant/Formulation 1/Proof 3 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1/Proof_3 | [
"Value of Vandermonde Determinant"
] | [
"Definition:Vandermonde Determinant/Formulation 1"
] | [
"Square Matrix with Duplicate Rows has Zero Determinant",
"Polynomial Factor Theorem"
] |
proofwiki-18607 | Value of Vandermonde Determinant/Formulation 1 | Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin {vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_n & {x_n}^2 & \cdots & {x_n}^{n - 2} & {x_n}^{n - 1}
\end {vm... | Let $V_n$ be the '''[[Definition:Vandermonde Determinant/Formulation 1|Vandermonde determinant of order $n$]]''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 1}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | Let:
:<nowiki>$V_n = \begin {vmatrix}
1 & x_1 & {x_1}^2 & \cdots & {x_1}^{n - 2} & {x_1}^{n - 1} \\
1 & x_2 & {x_2}^2 & \cdots & {x_2}^{n - 2} & {x_2}^{n - 1} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_n & {x_n}^2 & \cdots & {x_n}^{n - 2} & {x_n}^{n - 1}
\end {v... | Value of Vandermonde Determinant/Formulation 1/Proof 4 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_1/Proof_4 | [
"Value of Vandermonde Determinant"
] | [
"Definition:Vandermonde Determinant/Formulation 1"
] | [
"Definition:Monic Polynomial",
"Definition:Degree of Polynomial",
"Effect of Elementary Row Operations on Determinant",
"Laplace Expansion Theorem for Determinants",
"Definition:Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Step",
"Value of Vandermonde Determinan... |
proofwiki-18608 | Value of Vandermonde Determinant/Formulation 2 | Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 2}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le j \mathop \le n} x_j \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$ | The proof follows directly from that for Value of Vandermonde Determinant/Formulation 1 and the result Determinant with Row Multiplied by Constant.
{{finish}}
{{Namedfor|Alexandre-Théophile Vandermonde}} | Let $V_n$ be the '''[[Definition:Vandermonde Determinant/Formulation 2|Vandermonde determinant of order $n$]]''' defined as the following formulation:
{{:Definition:Vandermonde Determinant/Formulation 2}}
Its value is given by:
:$\ds V_n = \prod_{1 \mathop \le j \mathop \le n} x_j \prod_{1 \mathop \le i \mathop < j \m... | The proof follows directly from that for [[Value of Vandermonde Determinant/Formulation 1]] and the result [[Determinant with Row Multiplied by Constant]].
{{finish}}
{{Namedfor|Alexandre-Théophile Vandermonde}} | Value of Vandermonde Determinant/Formulation 2 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_2 | https://proofwiki.org/wiki/Value_of_Vandermonde_Determinant/Formulation_2 | [
"Value of Vandermonde Determinant"
] | [
"Definition:Vandermonde Determinant/Formulation 2"
] | [
"Value of Vandermonde Determinant/Formulation 1",
"Determinant with Row Multiplied by Constant"
] |
proofwiki-18609 | Quantity of Positive Integers Divisible by Particular Integer | Let $d$ be a positive integer.
Let $x \ge 1$ be a real number.
Then:
:$\ds \sum_{n \le x, \, d \divides n} 1 = \floor {\frac x d}$
That is:
:there are $\floor {\dfrac x d}$ natural numbers less than or equal to $x$ that are divisible by $d$. | Consider the sum:
:$\ds \sum_{n \le x, \, d \divides n} 1$
Note that a natural number $n \le x$ is divisible by $d$ {{iff}}:
:there exists a natural number $k$ such that $n = d k$.
So we are counting the natural numbers $k$ such that $d k \le x$.
That is, the natural numbers $k$ such that:
:$k \le \dfrac x d$
So:
{... | Let $d$ be a [[Definition:Positive Integer|positive integer]].
Let $x \ge 1$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \sum_{n \le x, \, d \divides n} 1 = \floor {\frac x d}$
That is:
:there are $\floor {\dfrac x d}$ [[Definition:Natural Number|natural numbers]] less than or equal to $x$ that ar... | Consider the sum:
:$\ds \sum_{n \le x, \, d \divides n} 1$
Note that a [[Definition:Natural Number|natural number]] $n \le x$ is [[Definition:Divisor of Integer|divisible]] by $d$ {{iff}}:
:there exists a [[Definition:Natural Number|natural number]] $k$ such that $n = d k$.
So we are counting the [[Definition:Natu... | Quantity of Positive Integers Divisible by Particular Integer | https://proofwiki.org/wiki/Quantity_of_Positive_Integers_Divisible_by_Particular_Integer | https://proofwiki.org/wiki/Quantity_of_Positive_Integers_Divisible_by_Particular_Integer | [
"Analytic Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Real Number",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Category:Analytic Number Theory"
] |
proofwiki-18610 | Equivalence of Definitions of P-adic Integer | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
{{TFAE|def=P-adic Integer}}
=== Definition 1 ===
{{:Definition:P-adic Integer/Definition 1}}
=== Definition 2 ===
{{:Definition:P-adic Integer/Definition 2}} | === Definition 1 implies Definition 2 ===
{{:Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2}}{{qed|lemma}} | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
{{TFAE|def=P-adic Integer}}
=== [[Definition:P-adic Integer/Definition 1|Definition 1]] ===
{{:Definition:P-adic Integer/Definition 1}}
=== [[Definition:P-a... | === [[Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2|Definition 1 implies Definition 2]] ===
{{:Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2}}{{qed|lemma}} | Equivalence of Definitions of P-adic Integer | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer | [
"P-adic Integers",
"Equivalence of Definitions of P-adic Integer"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer/Definition 1",
"Definition:P-adic Integer/Definition 2"
] | [
"Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2"
] |
proofwiki-18611 | Order of Sum of Reciprocal of Primes | :$\ds \sum_{p \mathop \le x} \frac 1 p = \map \ln {\ln x} + \map \OO 1$ | We have:
{{begin-eqn}}
{{eqn | l = \int_p^x \frac 1 {t \ln^2 t} \rd t
| r = \int_{\ln p}^{\ln x} \frac {e^u} {e^u u^2} \rd u
| c = substituting $t \mapsto e^u$
}}
{{eqn | r = \int_{\ln p}^{\ln x} \frac 1 {u^2} \rd u
}}
{{eqn | r = \intlimits {-\frac 1 u} {\ln p} {\ln x}
| c = Primitive of Power, Fundamental Theorem... | :$\ds \sum_{p \mathop \le x} \frac 1 p = \map \ln {\ln x} + \map \OO 1$ | We have:
{{begin-eqn}}
{{eqn | l = \int_p^x \frac 1 {t \ln^2 t} \rd t
| r = \int_{\ln p}^{\ln x} \frac {e^u} {e^u u^2} \rd u
| c = [[Integration by Substitution|substituting]] $t \mapsto e^u$
}}
{{eqn | r = \int_{\ln p}^{\ln x} \frac 1 {u^2} \rd u
}}
{{eqn | r = \intlimits {-\frac 1 u} {\ln p} {\ln x}
| c = [[Pri... | Order of Sum of Reciprocal of Primes | https://proofwiki.org/wiki/Order_of_Sum_of_Reciprocal_of_Primes | https://proofwiki.org/wiki/Order_of_Sum_of_Reciprocal_of_Primes | [
"Order of Sum of Reciprocal of Primes",
"Analytic Number Theory"
] | [] | [
"Integration by Substitution",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Integration by Substitution",
"Fundamental Theorem of Calculus"
] |
proofwiki-18612 | Similarity Mapping on Plane with Scale Factor Minus 1 | Let $s_{-1}: \R^2 \to \R^2$ be a similarity mapping on $\R^2$ whose scale factor is $-1$.
Then $s_{-1}$ is the same as the rotation $r_\pi$ of the plane about the origin one half turn. | Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.
Then:
{{begin-eqn}}
{{eqn | l = \map {r_\pi} P
| r = \tuple {\paren {\cos \pi - \sin \pi} x, \paren {\sin \pi + \cos \pi} y}
| c = Rotation of Plane about Origin is Linear Operator
}}
{{eqn | r = \tuple {\paren {\paren {-1} - 0} x, \paren ... | Let $s_{-1}: \R^2 \to \R^2$ be a [[Definition:Similarity Mapping|similarity mapping]] on $\R^2$ whose [[Definition:Scale Factor|scale factor]] is $-1$.
Then $s_{-1}$ is the same as the [[Definition:Plane Rotation|rotation]] $r_\pi$ of [[Definition:The Plane|the plane]] about the [[Definition:Origin|origin]] one [[Def... | Let $P = \tuple {x, y} \in \R^2$ be an aribtrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Then:
{{begin-eqn}}
{{eqn | l = \map {r_\pi} P
| r = \tuple {\paren {\cos \pi - \sin \pi} x, \paren {\sin \pi + \cos \pi} y}
| c = [[Rotation of Plane about Origin is Linear Operator]]
}}
{{e... | Similarity Mapping on Plane with Scale Factor Minus 1 | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_with_Scale_Factor_Minus_1 | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_with_Scale_Factor_Minus_1 | [
"Similarity Mappings",
"Geometric Rotations"
] | [
"Definition:Similarity Mapping",
"Definition:Similarity Mapping/Scale Factor",
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Half Turn"
] | [
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Rotation of Plane about Origin is Linear Operator",
"Cosine of Straight Angle",
"Sine of Straight Angle"
] |
proofwiki-18613 | Similarity Mapping on Plane with Negative Parameter | Let $\beta \in \R_{<0}$ be a (strictly) negative real number.
Let $s_\beta: \R^2 \to \R^2$ be the similarity mapping on $\R^2$ whose scale factor is $\beta$.
Then $s_\beta$ is a stretching or contraction followed by a rotation one half turn. | Let $\beta = -\gamma$ where $\gamma \in \R_{>0}$.
Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.
Then:
{{begin-eqn}}
{{eqn | l = \map {s_\beta} P
| r = \tuple {\paren {-\gamma} x, \paren {-\gamma} y}
| c = Definition of $\beta$
}}
{{eqn | r = \paren {-1} \tuple {\gamma x, \gamma y}
... | Let $\beta \in \R_{<0}$ be a [[Definition:Strictly Negative Real Number|(strictly) negative real number]].
Let $s_\beta: \R^2 \to \R^2$ be the [[Definition:Similarity Mapping|similarity mapping]] on $\R^2$ whose [[Definition:Scale Factor|scale factor]] is $\beta$.
Then $s_\beta$ is a [[Definition:Stretching|stretchi... | Let $\beta = -\gamma$ where $\gamma \in \R_{>0}$.
Let $P = \tuple {x, y} \in \R^2$ be an aribtrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Then:
{{begin-eqn}}
{{eqn | l = \map {s_\beta} P
| r = \tuple {\paren {-\gamma} x, \paren {-\gamma} y}
| c = Definition of $\beta$
}}
{{eqn ... | Similarity Mapping on Plane with Negative Parameter | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_with_Negative_Parameter | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_with_Negative_Parameter | [
"Similarity Mappings"
] | [
"Definition:Strictly Negative/Real Number",
"Definition:Similarity Mapping",
"Definition:Similarity Mapping/Scale Factor",
"Definition:Stretching",
"Definition:Contraction",
"Rotation of Plane about Origin is Linear Operator",
"Definition:Half Turn"
] | [
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Stretching",
"Definition:Contraction",
"Similarity Mapping on Plane with Scale Factor Minus 1",
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Angle",
"Definition:Half Turn",
"Definitio... |
proofwiki-18614 | Similarity Mapping on Plane Commutes with Half Turn about Origin | Let $\beta \in \R_{>0}$ be a (strictly) positive real number.
Let $s_{-\beta}: \R^2 \to \R^2$ be the similarity mapping on $\R^2$ whose scale factor is $-\beta$.
Then $s_{-\beta}$ is the same as:
:a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn
and:
:a rotation one half turn fol... | Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.
From Similarity Mapping on Plane with Negative Parameter, $s_{-\beta}$ is a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn.
Thus:
{{begin-eqn}}
{{eqn | l = \map {s_{-\beta} } P
| r = \map {s_{-1} } {\map {... | Let $\beta \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $s_{-\beta}: \R^2 \to \R^2$ be the [[Definition:Similarity Mapping|similarity mapping]] on $\R^2$ whose [[Definition:Scale Factor|scale factor]] is $-\beta$.
Then $s_{-\beta}$ is the same as:
:a [[Definitio... | Let $P = \tuple {x, y} \in \R^2$ be an aribtrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
From [[Similarity Mapping on Plane with Negative Parameter]], $s_{-\beta}$ is a [[Definition:Stretching|stretching]] or [[Definition:Contraction|contraction]] of [[Definition:Scale Factor|scale factor]] $... | Similarity Mapping on Plane Commutes with Half Turn about Origin | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_Commutes_with_Half_Turn_about_Origin | https://proofwiki.org/wiki/Similarity_Mapping_on_Plane_Commutes_with_Half_Turn_about_Origin | [
"Similarity Mappings",
"Geometric Rotations"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Similarity Mapping",
"Definition:Similarity Mapping/Scale Factor",
"Definition:Stretching",
"Definition:Contraction",
"Definition:Similarity Mapping/Scale Factor",
"Rotation of Plane about Origin is Linear Operator",
"Definition:Half Turn",
"De... | [
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Similarity Mapping on Plane with Negative Parameter",
"Definition:Stretching",
"Definition:Contraction",
"Definition:Similarity Mapping/Scale Factor",
"Rotation of Plane about Origin is Linear Operator",
"Definition:Half Turn",
"Similarity M... |
proofwiki-18615 | P-adic Integers is Valuation Ring Induced by P-adic Norm | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Then:
:the $p$-adic integers, $\Z_p$, is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$ | By the definition of the $p$-adic integers:
:$\Z_p = \set {x \in \Q_p : \norm x_p \le 1}$
From P-adic Numbers form Non-Archimedean Valued Field:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a non-Archimedean valued field.
By definition of the valuation ring induced by a non-Archimedean norm:
:$\Z_p$ is the valuation ring ... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Then:
:the [[Definition:P-adic Integer|$p$-adic integers]], $\Z_p$, is the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]... | By the definition of the [[Definition:P-adic Integer|$p$-adic integers]]:
:$\Z_p = \set {x \in \Q_p : \norm x_p \le 1}$
From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean valued field]].
By definition o... | P-adic Integers is Valuation Ring Induced by P-adic Norm | https://proofwiki.org/wiki/P-adic_Integers_is_Valuation_Ring_Induced_by_P-adic_Norm | https://proofwiki.org/wiki/P-adic_Integers_is_Valuation_Ring_Induced_by_P-adic_Norm | [
"P-adic Integers",
"P-adic Integers is Valuation Ring Induced by P-adic Norm"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:P-adic Integer",
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Valuation Ring Induced by Non-Archimedean Norm"
] |
proofwiki-18616 | Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2 | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$ such that $\norm x_p \le 1$.
Then the canonical expansion of $x$ contains only positive powers of $p$. | Let $x \in \Q_p$ such that $\norm x_p \le 1$.
From P-adic Integer is Limit of Unique P-adic Expansion, there exists a $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
By definition of the canonical expansion:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$ is the canonical expansion of $x$
It follo... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $x \in \Q_p$ such that $\norm x_p \le 1$.
Then the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$ contains only [[Definition:Positi... | Let $x \in \Q_p$ such that $\norm x_p \le 1$.
From [[P-adic Integer is Limit of Unique P-adic Expansion]], there exists a [[Definition:P-adic Expansion|$p$-adic expansion]] of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
By definition of the [[Definition:Canonical P-adic Expansion|canonical expansion]]:
:$\ds... | Equivalence of Definitions of P-adic Integer/Definition 1 Implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer/Definition_1_Implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer/Definition_1_Implies_Definition_2 | [
"Equivalence of Definitions of P-adic Integer"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Canonical P-adic Expansion",
"Definition:Positive",
"Definition:Power (Algebra)"
] | [
"P-adic Integer is Limit of Unique P-adic Expansion",
"Definition:P-adic Expansion",
"Definition:Canonical P-adic Expansion",
"Definition:Canonical P-adic Expansion",
"Definition:Canonical P-adic Expansion",
"Definition:Positive",
"Definition:Power (Algebra)"
] |
proofwiki-18617 | Equivalence of Definitions of P-adic Integer/Definition 2 Implies Definition 1 | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the valued field of $p$-adic numbers for some prime $p$.
That is, such that:
:$\Q_p$ is the field of $p$-adic numbers
:$\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q_p$.
Let $x \in \Q_p$ such that the canonical expansion of $x$ contains only positive powers of $p$.
Then:... | Let the canonical expansion of $x$ contain only positive powers of $p$.
That is:
:$x = \ds \sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N : 0 \le d_n < p$
==== Case 1 : $\forall n \in \N : d_n = 0$ ====
Let:
:$\forall n \in \N : d_n = 0$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sum_{n \mathop = 0}^\infty... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|valued field of $p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
That is, such that:
:$\Q_p$ is the [[Definition:Field of P-adic Numbers|field of $p$-adic numbers]]
:$\norm {\,\cdot\,}_p$ is the [[Definition... | Let the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$ contain only [[Definition:Positive|positive]] [[Definition:Power (Algebra)|powers]] of $p$.
That is:
:$x = \ds \sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N : 0 \le d_n < p$
==== Case 1 : $\forall n \in \N : d_n = 0$ ====
Let:
:$... | Equivalence of Definitions of P-adic Integer/Definition 2 Implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer/Definition_2_Implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Integer/Definition_2_Implies_Definition_1 | [
"Equivalence of Definitions of P-adic Integer"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Field of P-adic Numbers",
"Definition:P-adic Norm/P-adic Numbers",
"Definition:Canonical P-adic Expansion",
"Definition:Positive",
"Definition:Power (Algebra)"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Positive",
"Definition:Power (Algebra)",
"P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient"
] |
proofwiki-18618 | Sequence of P-adic Integers has Convergent Subsequence | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\sequence{x_n}$ be a sequence of $p$-adic integers.
Then:
:there exists a convergent subsequence $\sequence {x_{n_r} }_{r \mathop \in \N}$ of $\sequence{x_n}$ | === Lemma 1 ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 1}}{{qed|lemma}}
=== Lemma 2 ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 2}}{{qed|lemma}}
=== Lemma 3 ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 3}}{{qed|lemma}}
=== Lemma 4 ===
{{:Sequence o... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\sequence{x_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:there exists a [[Definition:Convergent ... | === [[Sequence of P-adic Integers has Convergent Subsequence/Lemma 1|Lemma 1]] ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 1}}{{qed|lemma}}
=== [[Sequence of P-adic Integers has Convergent Subsequence/Lemma 2|Lemma 2]] ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 2}}{{qe... | Sequence of P-adic Integers has Convergent Subsequence/Proof 1 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Proof_1 | [
"P-adic Integers",
"Sequence of P-adic Integers has Convergent Subsequence"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Sequence",
"Definition:P-adic Integer",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:Subsequence"
] | [
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 1",
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 2",
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 3",
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 4",
"P-adic Expansion is a Cauchy Sequence in P-... |
proofwiki-18619 | Sequence of P-adic Integers has Convergent Subsequence | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\sequence{x_n}$ be a sequence of $p$-adic integers.
Then:
:there exists a convergent subsequence $\sequence {x_{n_r} }_{r \mathop \in \N}$ of $\sequence{x_n}$ | From P-adic Integers are Compact Subspace:
:$\Z_p$ is a compact subspace in the metric space induced by $\norm{\,\cdot\,}_p$
From Compact Subspace of Metric Space is Sequentially Compact in Itself:
:$\Z_p$ is sequentially compact in itself
By definition of sequentially compact in itself:
:every sequence in $\Z_p$ has a... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\sequence{x_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:there exists a [[Definition:Convergent ... | From [[P-adic Integers are Compact Subspace]]:
:$\Z_p$ is a [[Definition:Compact Subspace|compact subspace]] in the [[Definition:Metric Space|metric space]] [[Definition:Metric Induced by Norm on Division Ring|induced]] by $\norm{\,\cdot\,}_p$
From [[Compact Subspace of Metric Space is Sequentially Compact in Itself]]... | Sequence of P-adic Integers has Convergent Subsequence/Proof 2 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Proof_2 | [
"P-adic Integers",
"Sequence of P-adic Integers has Convergent Subsequence"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Sequence",
"Definition:P-adic Integer",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:Subsequence"
] | [
"Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers",
"Definition:Compact Topological Space/Subspace",
"Definition:Metric Space",
"Definition:Metric Induced by Norm on Division Ring",
"Compact Subspace of Metric Space is Sequentially Compact in Itself",
"Definition:Sequentially ... |
proofwiki-18620 | P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary | {{begin-itemize}}
{{item|(a):|the $p$-adic integers, $\Z_p$, is a local ring}}
{{item|(b):|the principal ideal $p\Z_p$ is the unique maximal ideal of $\Z_p$}}
{{end-itemize}} | From $p$-adic Integers is Valuation Ring Induced by $p$-adic Norm:
:$\Z_p$ is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$
From Valuation Ideal of $p$-adic Numbers:
:$p \Z_p$ is the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$
From {{Corollary|Valuation Ideal... | {{begin-itemize}}
{{item|(a):|the [[Definition:P-adic Integer|$p$-adic integers]], $\Z_p$, is a [[Definition:Local Ring|local ring]]}}
{{item|(b):|the [[Definition:Principal Ideal|principal ideal]] $p\Z_p$ is the [[Definition:Unique|unique]] [[Definition:Maximal Ideal|maximal ideal]] of $\Z_p$}}
{{end-itemize}} | From [[P-adic Integers is Valuation Ring Induced by P-adic Norm|$p$-adic Integers is Valuation Ring Induced by $p$-adic Norm]]:
:$\Z_p$ is the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdo... | P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary | https://proofwiki.org/wiki/P-adic_Integers_is_Valuation_Ring_Induced_by_P-adic_Norm/Corollary | https://proofwiki.org/wiki/P-adic_Integers_is_Valuation_Ring_Induced_by_P-adic_Norm/Corollary | [
"P-adic Integers is Valuation Ring Induced by P-adic Norm"
] | [
"Definition:P-adic Integer",
"Definition:Local Ring",
"Definition:Principal Ideal",
"Definition:Unique",
"Definition:Maximal Ideal"
] | [
"P-adic Integers is Valuation Ring Induced by P-adic Norm",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Valuation Ideal of P-adic Numbers",
"Definition:Valuation Ideal Induced by Non-Archimedean Norm",
"Definition:Non-Archimedean/Norm (D... |
proofwiki-18621 | Equations defining Plane Reflection/Cartesian | Let $\LL$ be a straight line through the origin $O$ of a cartesian plane.
Let the angle between $\LL$ and the $x$-axis be $\alpha$.
Let $\phi_\alpha$ denote the reflection in the plane whose axis is $\LL$.
Let $P = \tuple {x, y}$ be an arbitrary point in the plane.
Then:
:$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha... | :420px
Let $\LL$ reflect $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.
Let $OP$ form an angle $\theta$ with the $x$-axis.
We have:
:$OP = OP'$
Thus:
{{begin-eqn}}
{{eqn | l = x
| r = OP \cos \theta
}}
{{eqn | l = y
| r = OP \sin \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x'
| r = OP \map... | Let $\LL$ be a [[Definition:Infinite Straight Line|straight line]] through the [[Definition:Origin|origin]] $O$ of a [[Definition:Cartesian Plane|cartesian plane]].
Let the [[Definition:Plane Angle|angle]] between $\LL$ and the [[Definition:X-Axis|$x$-axis]] be $\alpha$.
Let $\phi_\alpha$ denote the [[Definition:Plan... | :[[File:Reflection-equations-origin.png|420px]]
Let $\LL$ [[Definition:Plane Reflection|reflect]] $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.
Let $OP$ form an [[Definition:Plane Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
We have:
:$OP = OP'$
Thus:
{{begin-eqn}}
{{eqn | l = x
| r = O... | Equations defining Plane Reflection/Cartesian | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Cartesian | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Cartesian | [
"Geometric Reflections",
"Equations defining Plane Reflection"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Reflection (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Reflection (Geometry)/Plane/Axis",
"Definition:Poin... | [
"File:Reflection-equations-origin.png",
"Definition:Reflection (Geometry)/Plane",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cosine of Difference",
"Sine of Difference"
] |
proofwiki-18622 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 3 | :there exists a sequence $\sequence{b_n}$ of $p$-adic digits:
::for all $j \in \N$, there exists infinitely many $n \in \N$ such that the canonical expansion of $x_n$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$ | === Lemma 1 ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 1}}{{qed|lemma}} | :there exists a [[Definition:Sequence|sequence]] $\sequence{b_n}$ of [[Definition:P-adic Digit|$p$-adic digits]]:
::for all $j \in \N$, there exists [[Definition:Infinite|infinitely many]] $n \in \N$ such that the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x_n$ begins with the [[Definition:P-adic... | === [[Sequence of P-adic Integers has Convergent Subsequence/Lemma 1|Lemma 1]] ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 1}}{{qed|lemma}} | Sequence of P-adic Integers has Convergent Subsequence/Lemma 3 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_3 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_3 | [] | [
"Definition:Sequence",
"Definition:P-adic Digit",
"Definition:Infinite",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] | [
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 1",
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 1"
] |
proofwiki-18623 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 4 | :there exists a subsequence $\sequence{x_{n_j}}_{j \mathop \in \N}$ of $\sequence{x_n}$:
::for all $j \in \N$, the canonical expansion of $x_{n_j}$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$ | The proof proceeds using the principle of recursive definition.
For any non-empty subset $S$ of $\N$, let $\min S$ denote the smallest element of $S$.
From the Well-Ordering Principle, for any non-empty subset $S$ of $\N$, $\min S$ always exists.
Let $T = \N \times \N$.
Let $n_0 = \min \set{n \in \N : \text{ the canoni... | :there exists a [[Definition:Subsequence|subsequence]] $\sequence{x_{n_j}}_{j \mathop \in \N}$ of $\sequence{x_n}$:
::for all $j \in \N$, the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x_{n_j}$ begins with the [[Definition:P-adic Digit|$p$-adic digits]] $b_j \, \ldots \, b_1 b_0$ | The proof proceeds using the [[Principle of Recursive Definition|principle of recursive definition]].
For any [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $S$ of $\N$, let $\min S$ denote the [[Definition:Smallest Element|smallest element]] of $S$.
From the [[Well-Ordering Principle]], for any ... | Sequence of P-adic Integers has Convergent Subsequence/Lemma 4 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_4 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_4 | [] | [
"Definition:Subsequence",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] | [
"Principle of Recursive Definition",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Smallest Element",
"Well-Ordering Principle",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Hypothesis",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Canonica... |
proofwiki-18624 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 5 | :the subsequence $\sequence{x_{n_j}}_{j \mathop \in \N}$ converges to $x \in \Z_p$ | By definition of the canonical expansion $\ldots \, b_n \, \ldots \, b_1 b_0$ converges to $x$:
:the sequence of partial sums $\ds \sum_{n \mathop = 0}^j b_n p^n$ converges to $x$
Let $\sequence{y_j}$ be the sequence of partial sums:
:$y_j = \ds \sum_{n \mathop = m}^j b_n p^n$
From Null Sequence Test for Convergence, i... | :the [[Definition:Subsequence|subsequence]] $\sequence{x_{n_j}}_{j \mathop \in \N}$ [[Definition:Convergent P-adic Sequence|converges]] to $x \in \Z_p$ | By definition of the [[Definition:Canonical P-adic Expansion|canonical expansion]] $\ldots \, b_n \, \ldots \, b_1 b_0$ [[Definition:Convergent P-adic Sequence|converges]] to $x$:
:the [[Definition:Sequence|sequence]] of [[Definition:Partial Sum|partial sums]] $\ds \sum_{n \mathop = 0}^j b_n p^n$ [[Definition:Convergen... | Sequence of P-adic Integers has Convergent Subsequence/Lemma 5 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_5 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_5 | [] | [
"Definition:Subsequence",
"Definition:Convergent Sequence/P-adic Numbers"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:Sequence",
"Definition:Series/Sequence of Partial Sums",
"Definition:Convergent Sequence/P-adic Numbers",
"Definition:Sequence",
"Definition:Series/Sequence of Partial Sums",
"Null Sequence Test for C... |
proofwiki-18625 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 2 | Let $\sequence{b_0, b_1, \ldots, b_j}$ be a finite sequence of $p$-adic digits such that:
:there exists infinitely many $n \in \N$ such that the canonical expansion of $x_n$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$
Then there exists a $p$-adic digit $b_{j + 1}$ such that:
:there exists infinitely many... | === Lemma ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 6}}{{qed|lemma}} | Let $\sequence{b_0, b_1, \ldots, b_j}$ be a [[Definition:Finite Sequence|finite sequence]] of [[Definition:P-adic Digit|$p$-adic digits]] such that:
:there exists [[Definition:Infinite|infinitely many]] $n \in \N$ such that the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x_n$ begins with the [[Def... | === [[Sequence of P-adic Integers has Convergent Subsequence/Lemma 6|Lemma]] ===
{{:Sequence of P-adic Integers has Convergent Subsequence/Lemma 6}}{{qed|lemma}} | Sequence of P-adic Integers has Convergent Subsequence/Lemma 2 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_2 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_2 | [] | [
"Definition:Finite Sequence",
"Definition:P-adic Digit",
"Definition:Infinite",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit",
"Definition:P-adic Digit",
"Definition:Infinite",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] | [
"Sequence of P-adic Integers has Convergent Subsequence/Lemma 6"
] |
proofwiki-18626 | Absolute Value of Measurable Function is Measurable | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f : X \to \overline \R$ be a $\Sigma$-measurable function.
Then:
:$\size f$ is a $\Sigma$-measurable function. | From Characterization of Measurable Functions, it suffices to show that for each real number $t \in \R$, we have:
:$\set {x \in X : \size {\map f x} \le t} \in \Sigma$
If $t < 0$, we have:
:$\set {x \in X : \size {\map f x} \le t} = \O$
So, from Properties of Algebras of Sets, we have:
:$\set {x \in X : \size {\map ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f : X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Then:
:$\size f$ is a [[Definition:Measurable Function|$\Sigma$-measurable function]]. | From [[Characterization of Measurable Functions]], it suffices to show that for each [[Definition:Real Number|real number]] $t \in \R$, we have:
:$\set {x \in X : \size {\map f x} \le t} \in \Sigma$
If $t < 0$, we have:
:$\set {x \in X : \size {\map f x} \le t} = \O$
So, from [[Properties of Algebras of Sets]], ... | Absolute Value of Measurable Function is Measurable | https://proofwiki.org/wiki/Absolute_Value_of_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Absolute_Value_of_Measurable_Function_is_Measurable | [
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Measurable Function",
"Definition:Measurable Function"
] | [
"Characterization of Measurable Functions",
"Definition:Real Number",
"Properties of Algebras of Sets",
"Definition:Measurable Function",
"Characterization of Measurable Functions",
"Properties of Algebras of Sets",
"Definition:Intersection",
"Category:Measurable Functions"
] |
proofwiki-18627 | Monotone Convergence Theorem for Positive Simple Functions | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \R$ be a positive simple function.
Let $\EE^+$ be the space of positive simple functions.
For each $n \in \N$, let $f_n : X \to \R$ be a positive simple function, such that:
:$\ds \lim_{n \mathop \to \infty} f_n = f$
and:
:for each $x \in X$, the sequence... | Note that since:
:for each $x \in X$, the sequence $\sequence {\map {f_n} x}$ is increasing
we have that:
:$f_i \le f_j$
whenever $i \le j$.
Since $f_n \to f$, from Monotone Convergence Theorem (Real Analysis): Increasing Sequence, we further obtain:
:$f_i \le f_j \le f$
whenever $i \le j$.
From Integral of Posit... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f: X \to \R$ be a [[Definition:Positive Simple Function|positive simple function]].
Let $\EE^+$ be the [[Definition:Space of Positive Simple Functions|space of positive simple functions]].
For each $n \in \N$, let $f_n : X \to \R$ b... | Note that since:
:for each $x \in X$, the [[Definition:Sequence|sequence]] $\sequence {\map {f_n} x}$ is [[Definition:Increasing Sequence|increasing]]
we have that:
:$f_i \le f_j$
whenever $i \le j$.
Since $f_n \to f$, from [[Monotone Convergence Theorem (Real Analysis)/Increasing Sequence|Monotone Convergence... | Monotone Convergence Theorem for Positive Simple Functions | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_for_Positive_Simple_Functions | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_for_Positive_Simple_Functions | [
"Measure Theory"
] | [
"Definition:Measure Space",
"Definition:Simple Function",
"Definition:Space of Simple Functions",
"Definition:Simple Function",
"Definition:Sequence",
"Definition:Increasing/Sequence",
"Definition:Pointwise Limit",
"Definition:Integral Sign",
"Definition:Integral of Positive Measurable Function"
] | [
"Definition:Sequence",
"Definition:Increasing/Sequence",
"Monotone Convergence Theorem (Real Analysis)/Increasing Sequence",
"Integral of Positive Simple Function is Increasing",
"Definition:Sequence",
"Definition:Increasing/Sequence",
"Definition:Bounded Sequence",
"Monotone Convergence Theorem (Real... |
proofwiki-18628 | Pointwise Maximum of Simple Functions is Simple | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f, g : X \to \R$ be simple functions.
Then the pointwise maximum $\max \set {f, g}: X \to \R$ is also simple function. | From Pointwise Sum of Simple Functions is Simple Function:
:$f + g$ is simple.
From Scalar Multiple of Simple Function is Simple Function:
:$-g$ is simple.
Then, from Pointwise Sum of Simple Functions is Simple Function, we have:
:$f - g$ is simple.
From Absolute Value of Simple Function is Simple Function:
:$\size {f... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f, g : X \to \R$ be [[Definition:Simple Function|simple functions]].
Then the [[Definition:Pointwise Maximum of Extended Real-Valued Functions|pointwise maximum]] $\max \set {f, g}: X \to \R$ is also [[Definition:Simple Function|s... | From [[Pointwise Sum of Simple Functions is Simple Function]]:
:$f + g$ is [[Definition:Simple Function|simple]].
From [[Scalar Multiple of Simple Function is Simple Function]]:
:$-g$ is [[Definition:Simple Function|simple]].
Then, from [[Pointwise Sum of Simple Functions is Simple Function]], we have:
:$f - g$ i... | Pointwise Maximum of Simple Functions is Simple/Proof 1 | https://proofwiki.org/wiki/Pointwise_Maximum_of_Simple_Functions_is_Simple | https://proofwiki.org/wiki/Pointwise_Maximum_of_Simple_Functions_is_Simple/Proof_1 | [
"Simple Functions",
"Pointwise Maximum of Simple Functions is Simple"
] | [
"Definition:Measurable Space",
"Definition:Simple Function",
"Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions",
"Definition:Simple Function"
] | [
"Pointwise Sum of Simple Functions is Simple Function",
"Definition:Simple Function",
"Scalar Multiple of Simple Function is Simple Function",
"Definition:Simple Function",
"Pointwise Sum of Simple Functions is Simple Function",
"Definition:Simple Function",
"Absolute Value of Simple Function is Simple ... |
proofwiki-18629 | Pointwise Maximum of Simple Functions is Simple | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f, g : X \to \R$ be simple functions.
Then the pointwise maximum $\max \set {f, g}: X \to \R$ is also simple function. | From Simple Function is Measurable, we have that:
:$f$ and $g$ are $\Sigma$-measurable.
For brevity let:
:$h = \max \set {f, g}$
From Pointwise Maximum of Measurable Functions is Measurable, we have that:
:$h$ is $\Sigma$-measurable.
From Measurable Function is Simple Function iff Finite Image Set, we aim to show tha... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f, g : X \to \R$ be [[Definition:Simple Function|simple functions]].
Then the [[Definition:Pointwise Maximum of Extended Real-Valued Functions|pointwise maximum]] $\max \set {f, g}: X \to \R$ is also [[Definition:Simple Function|s... | From [[Simple Function is Measurable]], we have that:
:$f$ and $g$ are [[Definition:Measurable Function|$\Sigma$-measurable]].
For brevity let:
:$h = \max \set {f, g}$
From [[Pointwise Maximum of Measurable Functions is Measurable]], we have that:
:$h$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
... | Pointwise Maximum of Simple Functions is Simple/Proof 2 | https://proofwiki.org/wiki/Pointwise_Maximum_of_Simple_Functions_is_Simple | https://proofwiki.org/wiki/Pointwise_Maximum_of_Simple_Functions_is_Simple/Proof_2 | [
"Simple Functions",
"Pointwise Maximum of Simple Functions is Simple"
] | [
"Definition:Measurable Space",
"Definition:Simple Function",
"Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions",
"Definition:Simple Function"
] | [
"Simple Function is Measurable",
"Definition:Measurable Function",
"Pointwise Maximum of Measurable Functions is Measurable",
"Definition:Measurable Function",
"Measurable Function is Simple Function iff Finite Image Set",
"Definition:Finite Set",
"Measurable Function is Simple Function iff Finite Image... |
proofwiki-18630 | Equations defining Plane Reflection/Examples/X-Axis | Let $\phi_x$ denote the reflection in the plane whose axis is the $x$-axis.
Let $P = \tuple {x, y}$ be an arbitrary point in the plane
Then:
:$\map {\phi_x} P = \tuple {x, -y}$ | From Equations defining Plane Reflection:
:$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$
where $\alpha$ denotes the angle between the axis and the $x$-axis.
By definition, the $x$-axis, being coincident with itself, is at a zero angle with itself.
Hence $\phi_x$ ... | Let $\phi_x$ denote the [[Definition:Plane Reflection|reflection]] in [[Definition:The Plane|the plane]] whose [[Definition:Axis of Reflection|axis]] is the [[Definition:X-Axis|$x$-axis]].
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]]
Then:
:$\map {\phi_x} P... | From [[Equations defining Plane Reflection/Cartesian|Equations defining Plane Reflection]]:
:$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$
where $\alpha$ denotes the [[Definition:Plane Angle|angle]] between the [[Definition:Axis of Reflection|axis]] and the [[Def... | Equations defining Plane Reflection/Examples/X-Axis | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Examples/X-Axis | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Examples/X-Axis | [
"Equations defining Plane Reflection"
] | [
"Definition:Reflection (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Reflection (Geometry)/Plane/Axis",
"Definition:Axis/X-Axis",
"Definition:Point",
"Definition:Plane Surface/The Plane"
] | [
"Equations defining Plane Reflection/Cartesian",
"Definition:Angle",
"Definition:Reflection (Geometry)/Plane/Axis",
"Definition:Axis/X-Axis",
"Definition:Axis/X-Axis",
"Definition:Coincident Straight Lines",
"Definition:Zero Angle",
"Cosine of Zero is One",
"Sine of Zero is Zero"
] |
proofwiki-18631 | Equations defining Plane Reflection/Examples/Y-Axis | Let $\phi_y$ denote the reflection in the plane whose axis is the $y$-axis.
Let $P = \tuple {x, y}$ be an arbitrary point in the plane
Then:
:$\map {\phi_y} P = \tuple {-x, y}$ | From Equations defining Plane Reflection:
:$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$
where $\alpha$ denotes the angle between the axis and the $x$-axis.
By definition, the $y$-axis, is perpendicular to the $x$-axis
Hence $\phi_y$ can be expressed as $\phi_\al... | Let $\phi_y$ denote the [[Definition:Plane Reflection|reflection]] in [[Definition:The Plane|the plane]] whose [[Definition:Axis of Reflection|axis]] is the [[Definition:Y-Axis|$y$-axis]].
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]]
Then:
:$\map {\phi_y} P... | From [[Equations defining Plane Reflection/Cartesian|Equations defining Plane Reflection]]:
:$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$
where $\alpha$ denotes the [[Definition:Plane Angle|angle]] between the [[Definition:Axis of Reflection|axis]] and the [[Def... | Equations defining Plane Reflection/Examples/Y-Axis | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Examples/Y-Axis | https://proofwiki.org/wiki/Equations_defining_Plane_Reflection/Examples/Y-Axis | [
"Equations defining Plane Reflection"
] | [
"Definition:Reflection (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Reflection (Geometry)/Plane/Axis",
"Definition:Axis/Y-Axis",
"Definition:Point",
"Definition:Plane Surface/The Plane"
] | [
"Equations defining Plane Reflection/Cartesian",
"Definition:Angle",
"Definition:Reflection (Geometry)/Plane/Axis",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Angular Measure/Radian",
"Cosine of Straight Angle",
... |
proofwiki-18632 | Plane Reflection is Space Rotation | Let $M$ be a straight line in the plane passing through the origin.
Let $s_M$ be the '''reflection''' of $\R^2$ in $M$.
Then $s_M$ is the rotation of the plane in space through one half turn about $M$ as an axis. | {{ProofWanted|Needs equations of space rotation}} | Let $M$ be a [[Definition:Infinite Line|straight line]] in [[Definition:The Plane|the plane]] passing through the [[Definition:Origin|origin]].
Let $s_M$ be the '''[[Definition:Plane Reflection|reflection]]''' of $\R^2$ in $M$.
Then $s_M$ is the [[Definition:Space Rotation|rotation]] of [[Definition:The Plane|the pl... | {{ProofWanted|Needs equations of space rotation}} | Plane Reflection is Space Rotation | https://proofwiki.org/wiki/Plane_Reflection_is_Space_Rotation | https://proofwiki.org/wiki/Plane_Reflection_is_Space_Rotation | [
"Geometric Reflections",
"Geometric Rotations"
] | [
"Definition:Line/Infinite",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Reflection (Geometry)/Plane",
"Definition:Rotation (Geometry)/Space",
"Definition:Plane Surface/The Plane",
"Definition:Ordinary Space",
"Definition:Half Turn",
"Definition:Axis"
] | [] |
proofwiki-18633 | Plane Reflection is Involution | Let $M$ be a straight line in the plane passing through the origin.
Let $s_M$ be the '''reflection''' of $\R^2$ in $M$.
Then $s_M$ is an involution in the sense that:
:$s_M \circ s_M = I_{\R^2}$
where $I_{\R^2}$ is the identity mapping on $\R_2$.
That is:
:$s_M = {s_M}^{-1}$ | Let the angle between $M$ and the $x$-axis be $\alpha$.
Let $P = \tuple {x, y}$ be an arbitrary point in the plane.
Then from Equations defining Plane Reflection:
:$\map {s_M} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$
Thus:
{{begin-eqn}}
{{eqn | l = \map {s_M \circ s_M} P
... | Let $M$ be a [[Definition:Infinite Line|straight line]] in [[Definition:The Plane|the plane]] passing through the [[Definition:Origin|origin]].
Let $s_M$ be the '''[[Definition:Plane Reflection|reflection]]''' of $\R^2$ in $M$.
Then $s_M$ is an [[Definition:Involution (Mapping)|involution]] in the sense that:
:$s_M ... | Let the [[Definition:Plane Angle|angle]] between $M$ and the [[Definition:X-Axis|$x$-axis]] be $\alpha$.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Then from [[Equations defining Plane Reflection/Cartesian|Equations defining Plane Reflection]]:
:$\map {s... | Plane Reflection is Involution | https://proofwiki.org/wiki/Plane_Reflection_is_Involution | https://proofwiki.org/wiki/Plane_Reflection_is_Involution | [
"Geometric Reflections",
"Involutions"
] | [
"Definition:Line/Infinite",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Reflection (Geometry)/Plane",
"Definition:Involution (Mapping)",
"Definition:Identity Mapping"
] | [
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Equations defining Plane Reflection/Cartesian",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-18634 | Equations defining Plane Rotation/Cartesian | Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.
Let $P = \tuple {x, y}$ be an arbitrary point in the plane.
Then:
:$\map {r_\alpha} P = \tuple {x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha}$ | :420px
Let $r_\alpha$ rotate $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.
Let $OP$ form an angle $\theta$ with the $x$-axis.
We have:
:$OP = OP'$
Thus:
{{begin-eqn}}
{{eqn | l = x
| r = OP \cos \theta
}}
{{eqn | l = y
| r = OP \sin \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x'
| r = OP ... | Let $r_\alpha$ be the [[Definition:Plane Rotation|rotation]] of [[Definition:The Plane|the plane]] about the [[Definition:Origin|origin]] through an [[Definition:Angle|angle]] of $\alpha$.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Then:
:$\map {r_\alpha}... | :[[File:Rotation-equations-origin.png|420px]]
Let $r_\alpha$ [[Definition:Plane Rotation|rotate]] $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.
Let $OP$ form an [[Definition:Plane Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
We have:
:$OP = OP'$
Thus:
{{begin-eqn}}
{{eqn | l = x
| r = O... | Equations defining Plane Rotation/Cartesian | https://proofwiki.org/wiki/Equations_defining_Plane_Rotation/Cartesian | https://proofwiki.org/wiki/Equations_defining_Plane_Rotation/Cartesian | [
"Geometric Rotations",
"Equations defining Plane Rotation"
] | [
"Definition:Rotation (Geometry)/Plane",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Point",
"Definition:Plane Surface/The Plane"
] | [
"File:Rotation-equations-origin.png",
"Definition:Rotation (Geometry)/Plane",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cosine of Sum",
"Sine of Difference"
] |
proofwiki-18635 | Triangle Inequality for Integrals/Corollary | Let $f: X \to \overline \R$ be a $\mu$-integrable function be such that:
:$\ds \int \size f \rd \mu = 0$
Then:
:$\ds \int f \rd \mu = 0$ | From Triangle Inequality for Integrals, we have:
:$\ds \size {\int f \rd \mu} \le \int \size f \rd \mu$
We have:
:$\ds \int \size f \rd \mu = 0$
so:
:$\ds \size {\int f \rd \mu} \le 0$
That is:
:$\ds \size {\int f \rd \mu} = 0$
so:
:$\ds \int f \rd \mu = 0$
{{qed}}
Category:Triangle Inequality for Integrals
97byyxdw... | Let $f: X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]] be such that:
:$\ds \int \size f \rd \mu = 0$
Then:
:$\ds \int f \rd \mu = 0$ | From [[Triangle Inequality for Integrals]], we have:
:$\ds \size {\int f \rd \mu} \le \int \size f \rd \mu$
We have:
:$\ds \int \size f \rd \mu = 0$
so:
:$\ds \size {\int f \rd \mu} \le 0$
That is:
:$\ds \size {\int f \rd \mu} = 0$
so:
:$\ds \int f \rd \mu = 0$
{{qed}}
[[Category:Triangle Inequality for In... | Triangle Inequality for Integrals/Corollary | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Corollary | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Corollary | [
"Triangle Inequality for Integrals"
] | [
"Definition:Integrable Function/Measure Space"
] | [
"Triangle Inequality for Integrals",
"Category:Triangle Inequality for Integrals"
] |
proofwiki-18636 | Integral of Integrable Function over Measurable Set is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $E \in \Sigma$.
Let $f: X \to \overline \R$ be a $\mu$-integrable function.
Then the $\mu$-integral of $f$ over $E$:
:$\ds \int_E f \rd \mu = \int f \cdot \chi_E \rd \mu$
is well-defined. | We need to show that $f \cdot \chi_E$ is $\mu$-integrable, so that its $\mu$-integral is well-understood.
Since $f$ is $\mu$-integrable, it is certainly $\Sigma$-measurable.
From Characteristic Function Measurable iff Set Measurable, we have that:
:$\chi_E$ is $\Sigma$-measurable.
Then by Pointwise Product of Measurab... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $E \in \Sigma$.
Let $f: X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]].
Then the [[Definition:Integral of Measure-Integrable Function over Measurable Set|$\mu$-integral of $f$ over $E... | We need to show that $f \cdot \chi_E$ is [[Definition:Measure-Integrable Function|$\mu$-integrable]], so that its [[Definition:Integral of Measure-Integrable Function|$\mu$-integral]] is well-understood.
Since $f$ is [[Definition:Measure-Integrable Function|$\mu$-integrable]], it is certainly [[Definition:Measurable F... | Integral of Integrable Function over Measurable Set is Well-Defined | https://proofwiki.org/wiki/Integral_of_Integrable_Function_over_Measurable_Set_is_Well-Defined | https://proofwiki.org/wiki/Integral_of_Integrable_Function_over_Measurable_Set_is_Well-Defined | [
"Integrals of Measure-Integrable Functions"
] | [
"Definition:Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Integral of Measure-Integrable Function over Measurable Set"
] | [
"Definition:Integrable Function/Measure Space",
"Definition:Integral of Measure-Integrable Function",
"Definition:Integrable Function/Measure Space",
"Definition:Measurable Function",
"Characteristic Function Measurable iff Set Measurable",
"Definition:Measurable Function",
"Pointwise Product of Measura... |
proofwiki-18637 | Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $f : X \to \overline \R$ be a $\GG$-integrable function.
Suppose that, for all $G \in \GG$:
:$\ds \int_G f \rd \mu = 0$
Then $f = 0$ $\mu$-almost everywhere. | In view of Measurable Function Zero A.E. iff Absolute Value has Zero Integral, we shall show:
:$\ds \int \size f \rd \mu = 0$
Since $f$ is $\GG$-measurable:
:$G_+ := \set {x \in X : \map f x > 0} \in \GG$
and:
:$G_- := \set {x \in X : \map f x \le 0} \in \GG$
On the one hand:
{{begin-eqn}}
{{eqn | l = \int f^+ \rd \mu
... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\GG$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Let $f : X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\GG$-integrable function]].
Suppose that, for all $G \in \GG$:
:$\ds \int_G f... | In view of [[Measurable Function Zero A.E. iff Absolute Value has Zero Integral]], we shall show:
:$\ds \int \size f \rd \mu = 0$
Since $f$ is $\GG$-[[Definition:Measurable Set|measurable]]:
:$G_+ := \set {x \in X : \map f x > 0} \in \GG$
and:
:$G_- := \set {x \in X : \map f x \le 0} \in \GG$
On the one hand:
{{begin... | Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero/Proof 2 | https://proofwiki.org/wiki/Integrable_Function_with_Zero_Integral_on_Sub-Sigma-Algebra_is_A.E._Zero | https://proofwiki.org/wiki/Integrable_Function_with_Zero_Integral_on_Sub-Sigma-Algebra_is_A.E._Zero/Proof_2 | [
"Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero",
"Measure Theory"
] | [
"Definition:Measure Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Function/Measure Space",
"Definition:Almost Everywhere"
] | [
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral",
"Definition:Measurable Set",
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Integral of Measure-Integrable Function"
] |
proofwiki-18638 | Set of Points for which Measurable Function is Real-Valued is Measurable | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \overline \R$ be a $\Sigma$-measurable.
Then:
:$\set {x \in X : \map f x \in \R}$ is $\Sigma$-measurable. | Since $f$ is $\Sigma$-measurable, we have that:
:for all $n \in \N$ the set $\set {x \in X : \map f x \le n}$ is $\Sigma$-measurable
and:
:for all $n \in \N$ the set $\set {x \in X : -n \le \map f x}$ is $\Sigma$-measurable.
From $\sigma$-Algebra Closed under Countable Intersection, we have:
:$\set {x \in X : -n \le ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f: X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable]].
Then:
:$\set {x \in X : \map f x \in \R}$ is [[Definition:Measurable Set|$\Sigma$-measurable]]. | Since $f$ is [[Definition:Measurable Function|$\Sigma$-measurable]], we have that:
:for all $n \in \N$ the set $\set {x \in X : \map f x \le n}$ is [[Definition:Measurable Function|$\Sigma$-measurable]]
and:
:for all $n \in \N$ the set $\set {x \in X : -n \le \map f x}$ is [[Definition:Measurable Function|$\Sigma$-... | Set of Points for which Measurable Function is Real-Valued is Measurable | https://proofwiki.org/wiki/Set_of_Points_for_which_Measurable_Function_is_Real-Valued_is_Measurable | https://proofwiki.org/wiki/Set_of_Points_for_which_Measurable_Function_is_Real-Valued_is_Measurable | [
"Measurable Sets",
"Measurable Functions",
"Set of Points for which Measurable Function is Real-Valued is Measurable"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Measurable Set"
] | [
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Sigma-Algebra Closed under Countable Intersection",
"Definition:Measurable Function",
"Definition:Sigma-Algebra",
"Definition:Set Union/Countable Union",
"Definition:Measurable Function",
"Category... |
proofwiki-18639 | Set of Points for which Measurable Function is Real-Valued is Measurable/Corollary | :$\set {x \in X : \size {\map f x} = +\infty}$ is $\Sigma$-measurable. | We have:
:$\set {x \in X : \size {\map f x} = +\infty} = X \setminus \set {x \in X : \map f x \in \R}$
From Set of Points for which Measurable Function is Real-Valued is Measurable, we have:
:$\set {x \in X : \map f x \in \R}$ is $\Sigma$-measurable.
Since $\sigma$-algebras are closed under complementation, we have t... | :$\set {x \in X : \size {\map f x} = +\infty}$ is [[Definition:Measurable Set|$\Sigma$-measurable]]. | We have:
:$\set {x \in X : \size {\map f x} = +\infty} = X \setminus \set {x \in X : \map f x \in \R}$
From [[Set of Points for which Measurable Function is Real-Valued is Measurable]], we have:
:$\set {x \in X : \map f x \in \R}$ is [[Definition:Measurable Set|$\Sigma$-measurable]].
Since [[Definition:Sigma-Alge... | Set of Points for which Measurable Function is Real-Valued is Measurable/Corollary | https://proofwiki.org/wiki/Set_of_Points_for_which_Measurable_Function_is_Real-Valued_is_Measurable/Corollary | https://proofwiki.org/wiki/Set_of_Points_for_which_Measurable_Function_is_Real-Valued_is_Measurable/Corollary | [
"Set of Points for which Measurable Function is Real-Valued is Measurable"
] | [
"Definition:Measurable Set"
] | [
"Set of Points for which Measurable Function is Real-Valued is Measurable",
"Definition:Measurable Set",
"Definition:Sigma-Algebra",
"Definition:Relative Complement",
"Definition:Measurable Set",
"Category:Set of Points for which Measurable Function is Real-Valued is Measurable"
] |
proofwiki-18640 | Image of Projection in Plane | Let $M$ and $N$ be distinct lines in the plane.
:420px
Let $\pr_{M, N}$ be the '''projection on $M$ along $N$''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.
Then $M$ is the image of $\pr_{M, N}$. | Let $x \in \R^2$ be arbitrary.
By definition, the image of $x$ is the intersection of $M$ with the line through $x$ parallel to $N$.
Therefore $\map {\pr_{M, N} } x \in M$.
Hence:
:$\Img {\pr_{M, N} } \subseteq M$.
Now consider $y \in M$.
By Playfair's axiom there exists exactly one straight line $L$ parallel to $N$ pa... | Let $M$ and $N$ be distinct [[Definition:Straight Line|lines]] in [[Definition:The Plane|the plane]].
:[[File:Projection-in-plane.png|420px]]
Let $\pr_{M, N}$ be the '''[[Definition:Projection in Plane|projection on $M$ along $N$]]''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the [[Definition:Intersection (Geome... | Let $x \in \R^2$ be arbitrary.
By definition, the [[Definition:Image of Element under Mapping|image]] of $x$ is the [[Definition:Intersection (Geometry)|intersection]] of $M$ with the line through $x$ [[Definition:Parallel Lines|parallel]] to $N$.
Therefore $\map {\pr_{M, N} } x \in M$.
Hence:
:$\Img {\pr_{M, N} } \... | Image of Projection in Plane | https://proofwiki.org/wiki/Image_of_Projection_in_Plane | https://proofwiki.org/wiki/Image_of_Projection_in_Plane | [
"Geometric Projections"
] | [
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"File:Projection-in-plane.png",
"Definition:Projection (Geometry)/Plane",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Axiom:Playfair's Axiom",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Point",
"Defi... |
proofwiki-18641 | Kernel of Projection in Plane between Lines passing through Origin | Let $M$ and $N$ be distinct lines in the plane both of which pass through the origin $O$.
Let $\pr_{M, N}$ be the '''projection on $M$ along $N$''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.
Then $N$ is the kernel of $\pr_{M, N}$.
{{explain|As the k... | Let $\LL$ be the straight line through $x$ which is parallel to $N$.
Let $\map {\pr_{M, N} } x = \tuple {0, 0}$.
By definition, $\map {\pr_{M, N} } x$ is the intersection of $M$ with $\LL$.
However, as $\map {\pr_{M, N} } x = \tuple {0, 0}$, it follows that $\LL$ is coincident with $N$.
Hence the result.
{{qed}} | Let $M$ and $N$ be distinct [[Definition:Straight Line|lines]] in [[Definition:The Plane|the plane]] both of which pass through the [[Definition:Origin|origin]] $O$.
Let $\pr_{M, N}$ be the '''[[Definition:Projection in Plane|projection on $M$ along $N$]]''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the [[Defini... | Let $\LL$ be the [[Definition:Straight Line|straight line]] through $x$ which is [[Definition:Parallel Lines|parallel]] to $N$.
Let $\map {\pr_{M, N} } x = \tuple {0, 0}$.
By definition, $\map {\pr_{M, N} } x$ is the [[Definition:Intersection (Geometry)|intersection]] of $M$ with $\LL$.
However, as $\map {\pr_{M, N}... | Kernel of Projection in Plane between Lines passing through Origin | https://proofwiki.org/wiki/Kernel_of_Projection_in_Plane_between_Lines_passing_through_Origin | https://proofwiki.org/wiki/Kernel_of_Projection_in_Plane_between_Lines_passing_through_Origin | [
"Geometric Projections"
] | [
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Projection (Geometry)/Plane",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Kernel of Linear Transformation"
] | [
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Intersection (Geometry)",
"Definition:Coincident Straight Lines"
] |
proofwiki-18642 | Fixed Points of Projection in Plane | Let $M$ and $N$ be distinct lines in the plane.
:420px
Let $\pr_{M, N}$ be the '''projection on $M$ along $N$''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.
Then $M$ is the set of fixed points of $\pr_{M, N}$ in the sense that:
:$x \in M$
{{iff}}:
:$... | === Sufficient Condition ===
Let $x \in M$.
Let $\LL$ be the straight line through $x$ which is parallel to $N$.
As $x \in M$ it follows that $x$ is on the intersection of $M$ with $\LL$.
Hence by definition:
:$\map {\pr_{M, N} } x = x$
{{qed|lemma}} | Let $M$ and $N$ be distinct [[Definition:Straight Line|lines]] in [[Definition:The Plane|the plane]].
:[[File:Projection-in-plane.png|420px]]
Let $\pr_{M, N}$ be the '''[[Definition:Projection in Plane|projection on $M$ along $N$]]''':
:$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the [[Definition:Intersection (Geome... | === Sufficient Condition ===
Let $x \in M$.
Let $\LL$ be the [[Definition:Straight Line|straight line]] through $x$ which is [[Definition:Parallel Lines|parallel]] to $N$.
As $x \in M$ it follows that $x$ is on the [[Definition:Intersection (Geometry)|intersection]] of $M$ with $\LL$.
Hence by definition:
:$\map {\... | Fixed Points of Projection in Plane | https://proofwiki.org/wiki/Fixed_Points_of_Projection_in_Plane | https://proofwiki.org/wiki/Fixed_Points_of_Projection_in_Plane | [
"Geometric Projections"
] | [
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"File:Projection-in-plane.png",
"Definition:Projection (Geometry)/Plane",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Set",
"Definition:Fixed Point"
] | [
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Intersection (Geometry)",
"Definition:Intersection (Geometry)"
] |
proofwiki-18643 | Projection in Plane on X-Axis along Y-Axis | Let $\pr_{X, Y}$ denote the '''projection on the $x$-axis along the $y$-axis''':
:$\forall P \in \R^2: \map {\pr_{X, Y} } P =$ the intersection of the $x$-axis with the line through $P$ parallel to the $y$-axis.
Let $P = \tuple {\lambda_1, \lambda_2}$ be an arbitrary point in $\R^2$.
Then:
:$\map {\pr_{X, Y} } {\lambda... | This is an instance of a '''projection on $M$ along $N$''' where $N$ coincides with the $y$-axis.
Hence it is one of the special cases of Equations defining Projection in Plane: Cartesian:
:$\map {\pr_{M, N} } P = \tuple {\lambda_1, \lambda_1 \tan \theta}$
where $\theta$ is the angle between $M$ and the $x$-axis.
In th... | Let $\pr_{X, Y}$ denote the '''[[Definition:Projection in Plane|projection]] on the [[Definition:X-Axis|$x$-axis]] along the [[Definition:Y-Axis|$y$-axis]]''':
:$\forall P \in \R^2: \map {\pr_{X, Y} } P =$ the [[Definition:Set Intersection|intersection]] of the [[Definition:X-Axis|$x$-axis]] with the line through $P$ [... | This is an instance of a '''[[Definition:Projection in Plane|projection]] on $M$ along $N$''' where $N$ [[Definition:Coincident Straight Lines|coincides]] with the [[Definition:Y-Axis|$y$-axis]].
Hence it is one of the special cases of [[Equations defining Projection in Plane/Cartesian|Equations defining Projection in... | Projection in Plane on X-Axis along Y-Axis | https://proofwiki.org/wiki/Projection_in_Plane_on_X-Axis_along_Y-Axis | https://proofwiki.org/wiki/Projection_in_Plane_on_X-Axis_along_Y-Axis | [
"Geometric Projections"
] | [
"Definition:Projection (Geometry)/Plane",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Set Intersection",
"Definition:Axis/X-Axis",
"Definition:Parallel (Geometry)/Lines",
"Definition:Axis/Y-Axis",
"Definition:Point"
] | [
"Definition:Projection (Geometry)/Plane",
"Definition:Coincident Straight Lines",
"Definition:Axis/Y-Axis",
"Equations defining Projection in Plane/Cartesian",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Coincident Straight Lines",
"Definition:Axis/X-Axis",
"Tangent of Zero"
] |
proofwiki-18644 | Projection in Plane on Y-Axis along X-Axis | Let $\pr_{Y, X}$ denote the '''projection on the $y$-axis along the $x$-axis''':
:$\forall P \in \R^2: \map {\pr_{Y, X} } P =$ the intersection of the $y$-axis with the line through $P$ parallel to the $x$-axis.
Let $P = \tuple {\lambda_1, \lambda_2}$ be an arbitrary point in $\R^2$.
Then:
:$\map {\pr_{Y, X} } {\lambda... | This is an instance of a '''projection on $M$ along $N$''' where $M$ coincides with the $y$-axis.
Hence it is one of the special cases of Equations defining Projection in Plane: Cartesian:
:$\map {\pr_{M, N} } P = \tuple {0, \lambda_2 - \lambda_1 \tan \phi}$
where $\phi$ is the angle between $N$ and the $x$-axis.
In th... | Let $\pr_{Y, X}$ denote the '''[[Definition:Projection in Plane|projection]] on the [[Definition:Y-Axis|$y$-axis]] along the [[Definition:X-Axis|$x$-axis]]''':
:$\forall P \in \R^2: \map {\pr_{Y, X} } P =$ the [[Definition:Set Intersection|intersection]] of the [[Definition:Y-Axis|$y$-axis]] with the line through $P$ [... | This is an instance of a '''[[Definition:Projection in Plane|projection]] on $M$ along $N$''' where $M$ [[Definition:Coincident Straight Lines|coincides]] with the [[Definition:Y-Axis|$y$-axis]].
Hence it is one of the special cases of [[Equations defining Projection in Plane/Cartesian|Equations defining Projection in... | Projection in Plane on Y-Axis along X-Axis | https://proofwiki.org/wiki/Projection_in_Plane_on_Y-Axis_along_X-Axis | https://proofwiki.org/wiki/Projection_in_Plane_on_Y-Axis_along_X-Axis | [
"Geometric Projections"
] | [
"Definition:Projection (Geometry)/Plane",
"Definition:Axis/Y-Axis",
"Definition:Axis/X-Axis",
"Definition:Set Intersection",
"Definition:Axis/Y-Axis",
"Definition:Parallel (Geometry)/Lines",
"Definition:Axis/X-Axis",
"Definition:Point"
] | [
"Definition:Projection (Geometry)/Plane",
"Definition:Coincident Straight Lines",
"Definition:Axis/Y-Axis",
"Equations defining Projection in Plane/Cartesian",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Coincident Straight Lines",
"Definition:Axis/X-Axis",
"Tangent of Zero"
] |
proofwiki-18645 | Functions A.E. Equal iff Positive and Negative Parts A.E. Equal | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g : X \to \overline \R$ be extended real-valued functions.
Then $f = g$ $\mu$-almost everywhere {{iff}}:
:$f^+ = g^+$ $\mu$-almost everywhere and $f^- = g^-$ $\mu$-almost everywhere. | === Necessary Condition ===
Suppose that $f = g$ $\mu$-almost everywhere.
Then there exists a $\mu$-null set $N \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map g x$, then $x \in N$.
From the definition of the positive part, we have:
:$\map {f^+} x = \max \set {\map f x, 0}$
and:
:$\map {g^+} x = \max \s... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued functions]].
Then $f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]] {{iff}}:
:$f^+ = g^+$ [[Definition:Almost Everywhere... | === Necessary Condition ===
Suppose that $f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
Then there exists a [[Definition:Null Set|$\mu$-null set]] $N \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map g x$, then $x \in N$.
From the definition of the [[Definition:Positive Part|positive... | Functions A.E. Equal iff Positive and Negative Parts A.E. Equal | https://proofwiki.org/wiki/Functions_A.E._Equal_iff_Positive_and_Negative_Parts_A.E._Equal | https://proofwiki.org/wiki/Functions_A.E._Equal_iff_Positive_and_Negative_Parts_A.E._Equal | [
"Positive Parts",
"Negative Parts",
"Measure Theory",
"Positive Parts",
"Negative Parts"
] | [
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere"
] | [
"Definition:Almost Everywhere",
"Definition:Null Set",
"Definition:Positive Part",
"Proof by Contraposition",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Definition:Negative Part",
"Proof by Contraposition",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Definition:Almost Ev... |
proofwiki-18646 | A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 1 | Let $f: X \to \overline \R$ be a $\mu$-integrable function.
Let $g: X \to \overline \R$ be $\Sigma$-measurable.
Suppose that $f = g$ almost everywhere.
Then $g$ is also $\mu$-integrable, and:
:$\ds \int f \rd \mu = \int g \rd \mu$ | From Function Measurable iff Positive and Negative Parts Measurable, we have that:
:$g^+$, $f^+$, $g^-$ and $f^-$ are all $\Sigma$-measurable.
From Functions A.E. Equal iff Positive and Negative Parts A.E. Equal, we have that:
:$f^+ = g^+$ and $f^- = g^-$ $\mu$-almost everywhere.
Since $f^+$ and $g^+$ are positive $... | Let $f: X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]].
Let $g: X \to \overline \R$ be [[Definition:Measurable Function|$\Sigma$-measurable]].
Suppose that $f = g$ [[Definition:Almost Everywhere|almost everywhere]].
Then $g$ is also [[Definition:Measure-Integrable Func... | From [[Function Measurable iff Positive and Negative Parts Measurable]], we have that:
:$g^+$, $f^+$, $g^-$ and $f^-$ are all [[Definition:Measurable Function|$\Sigma$-measurable]].
From [[Functions A.E. Equal iff Positive and Negative Parts A.E. Equal]], we have that:
:$f^+ = g^+$ and $f^- = g^-$ [[Definition:Alm... | A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 1 | https://proofwiki.org/wiki/A.E._Equal_Positive_Measurable_Functions_have_Equal_Integrals/Corollary_1 | https://proofwiki.org/wiki/A.E._Equal_Positive_Measurable_Functions_have_Equal_Integrals/Corollary_1 | [
"A.E. Equal Positive Measurable Functions have Equal Integrals"
] | [
"Definition:Integrable Function/Measure Space",
"Definition:Measurable Function",
"Definition:Almost Everywhere",
"Definition:Integrable Function/Measure Space"
] | [
"Function Measurable iff Positive and Negative Parts Measurable",
"Definition:Measurable Function",
"Functions A.E. Equal iff Positive and Negative Parts A.E. Equal",
"Definition:Almost Everywhere",
"Definition:Measurable Function/Positive",
"Definition:Almost Everywhere",
"A.E. Equal Positive Measurabl... |
proofwiki-18647 | Constant Function is Measurable | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \overline \R$ be a constant extended real-valued function.
That is, there exists $c \in \overline \R$ such that:
:$\map f x = c$ for all $x \in X$.
Then $f$ is $\Sigma$-measurable. | By the definition of a $\Sigma$-measurable function, we aim to show that:
:$\set {x \in X : \map f x \le r}$ is $\Sigma$-measurable for each $r \in \R$.
First suppose that $\size c < \infty$.
Note that there are no $x \in X$ such that $\map f x < c$.
So for $r < c$, we have:
:$\set {x \in X : \map f x \le r} = \O$
F... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \overline \R$ be a [[Definition:Constant Mapping|constant]] [[Definition:Extended Real-Valued Function|extended real-valued function]].
That is, there exists $c \in \overline \R$ such that:
:$\map f x = c$ for all $x \in... | By the definition of a [[Definition:Measurable Function|$\Sigma$-measurable]] function, we aim to show that:
:$\set {x \in X : \map f x \le r}$ is [[Definition:Measurable Set|$\Sigma$-measurable]] for each $r \in \R$.
First suppose that $\size c < \infty$.
Note that there are no $x \in X$ such that $\map f x < c$.... | Constant Function is Measurable/Proof 1 | https://proofwiki.org/wiki/Constant_Function_is_Measurable | https://proofwiki.org/wiki/Constant_Function_is_Measurable/Proof_1 | [
"Constant Function is Measurable",
"Measurable Functions"
] | [
"Definition:Measure Space",
"Definition:Constant Mapping",
"Definition:Extended Real-Valued Function",
"Definition:Measurable Function"
] | [
"Definition:Measurable Function",
"Definition:Measurable Set",
"Sigma-Algebra Contains Empty Set",
"Definition:Sigma-Algebra",
"Sigma-Algebra Contains Empty Set",
"Definition:Sigma-Algebra"
] |
proofwiki-18648 | Constant Function is Measurable | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \overline \R$ be a constant extended real-valued function.
That is, there exists $c \in \overline \R$ such that:
:$\map f x = c$ for all $x \in X$.
Then $f$ is $\Sigma$-measurable. | First, suppose that $\size c < \infty$.
From Characteristic Function of Universe, we can write:
:$\map f x = c \map {\chi_X} x$
for each $x \in X$.
From the definition of a $\sigma$-algebra, we have:
:$X \in \Sigma$
So:
:$f$ is a simple function.
Then, from Simple Function is Measurable, we have:
:$f$ is $\Sigma$-... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \overline \R$ be a [[Definition:Constant Mapping|constant]] [[Definition:Extended Real-Valued Function|extended real-valued function]].
That is, there exists $c \in \overline \R$ such that:
:$\map f x = c$ for all $x \in... | First, suppose that $\size c < \infty$.
From [[Characteristic Function of Universe]], we can write:
:$\map f x = c \map {\chi_X} x$
for each $x \in X$.
From the definition of a [[Definition:Sigma-Algebra|$\sigma$-algebra]], we have:
:$X \in \Sigma$
So:
:$f$ is a [[Definition:Simple Function|simple function]]... | Constant Function is Measurable/Proof 2 | https://proofwiki.org/wiki/Constant_Function_is_Measurable | https://proofwiki.org/wiki/Constant_Function_is_Measurable/Proof_2 | [
"Constant Function is Measurable",
"Measurable Functions"
] | [
"Definition:Measure Space",
"Definition:Constant Mapping",
"Definition:Extended Real-Valued Function",
"Definition:Measurable Function"
] | [
"Characteristic Function of Universe",
"Definition:Sigma-Algebra",
"Definition:Simple Function",
"Simple Function is Measurable",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Pointwise Limit",
"Definition:Measurable Function",
"Pointwise Limit of Measurable Functio... |
proofwiki-18649 | Equations defining Projection in Plane/Cartesian | Let $M$ and $N$ be distinct lines in the plane, both of which pass through the origin.
Let the angle between $M$ and the $x$-axis be $\theta$.
Let the angle between $N$ and the $x$-axis be $\phi$.
Let $\pr_{M, N}$ be the '''projection on $M$ along $N$''':
:$\forall P \in \R^2: \map {\pr_{M, N} } P =$ the intersection o... | Let $P = \tuple {x_1, y_1} \in \R^2$ be arbitrary.
:540px
From Equation of Straight Line in Plane: Slope-Intercept Form, we can express $M$ and $N$ as follows:
{{begin-eqn}}
{{eqn | q = M
| l = y
| r = m x
}}
{{eqn | q = N
| l = y
| r = n x
}}
{{end-eqn}}
where:
{{begin-eqn}}
{{eqn | l = m
... | Let $M$ and $N$ be distinct [[Definition:Straight Line|lines]] in [[Definition:The Plane|the plane]], both of which pass through the [[Definition:Origin|origin]].
Let the [[Definition:Plane Angle|angle]] between $M$ and the [[Definition:X-Axis|$x$-axis]] be $\theta$.
Let the [[Definition:Plane Angle|angle]] between $... | Let $P = \tuple {x_1, y_1} \in \R^2$ be arbitrary.
:[[File:Projection-in-plane-equation.png|540px]]
From [[Equation of Straight Line in Plane/Slope-Intercept Form|Equation of Straight Line in Plane: Slope-Intercept Form]], we can express $M$ and $N$ as follows:
{{begin-eqn}}
{{eqn | q = M
| l = y
| r =... | Equations defining Projection in Plane/Cartesian | https://proofwiki.org/wiki/Equations_defining_Projection_in_Plane/Cartesian | https://proofwiki.org/wiki/Equations_defining_Projection_in_Plane/Cartesian | [
"Equations defining Projection in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Projection (Geometry)/Plane",
"Definition:Intersection (Geometry)",
"Definition:Para... | [
"File:Projection-in-plane-equation.png",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Mapping",
"Equation of Straight Line in Plane/Point-Slope Form"
] |
proofwiki-18650 | Bound for Positive Part of Pointwise Sum of Functions | Let $X$ be a set.
Let $f, g : X \to \overline \R$ be extended real-valued function.
Suppose that the pointwise sum $f + g$ is well-defined, that is:
:there exists no $x \in X$ such that $\set {\map f x, \map g x} = \set {\infty, -\infty}$.
Then:
:$\paren {f + g}^+ \le f^+ + g^+$
where $\paren {f + g}^+$, $f^+$ and $g... | Let $x \in X$.
From the definition of the positive part, we have:
:$\map {f^+} x = \max \set {\map f x, 0}$
and:
:$\map {g^+} x = \max \set {\map g x, 0}$
Suppose first that $\map f x$ and $\map g x$ are finite.
From Maximum Function in terms of Absolute Value, we then have:
:$\ds \map {f^+} x = \frac {\map f x + \... | Let $X$ be a [[Definition:Set|set]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued function]].
Suppose that the [[Definition:Pointwise Addition|pointwise sum]] $f + g$ is well-defined, that is:
:there exists no $x \in X$ such that $\set {\map f x, \map g x} = \se... | Let $x \in X$.
From the definition of the [[Definition:Positive Part|positive part]], we have:
:$\map {f^+} x = \max \set {\map f x, 0}$
and:
:$\map {g^+} x = \max \set {\map g x, 0}$
Suppose first that $\map f x$ and $\map g x$ are finite.
From [[Maximum Function in terms of Absolute Value]], we then have:
... | Bound for Positive Part of Pointwise Sum of Functions | https://proofwiki.org/wiki/Bound_for_Positive_Part_of_Pointwise_Sum_of_Functions | https://proofwiki.org/wiki/Bound_for_Positive_Part_of_Pointwise_Sum_of_Functions | [
"Measure Theory",
"Positive Parts",
"Positive Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Pointwise Addition",
"Definition:Positive Part"
] | [
"Definition:Positive Part",
"Maximum Function in terms of Absolute Value",
"Maximum Function in terms of Absolute Value",
"Triangle Inequality",
"Category:Positive Parts"
] |
proofwiki-18651 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 6 | there exists a subsequence $\sequence{y_n}$ of $\sequence{x_n}$:
:for all $n \in \N$, the canonical expansion of $y_n$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$ | For any non-empty subset $S$ of $\N$, let $\min S$ denote the smallest element of $S$.
From the Well-Ordering Principle, for any non-empty subset $S$ of $\N$, $\min S$ always exists.
Let $g:\N \to \N$ be the mapping defined by:
:$\map g n = \min \set{n > j : \text{ the canonical expansion of } x_n \text{ begins with th... | there exists a [[Definition:Subsequence|subsequence]] $\sequence{y_n}$ of $\sequence{x_n}$:
:for all $n \in \N$, the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $y_n$ begins with the [[Definition:P-adic Digit|$p$-adic digits]] $b_j \, \ldots \, b_1 b_0$ | For any [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $S$ of $\N$, let $\min S$ denote the [[Definition:Smallest Element|smallest element]] of $S$.
From the [[Well-Ordering Principle]], for any [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $S$ of $\N$, $\min S$ always exists... | Sequence of P-adic Integers has Convergent Subsequence/Lemma 6 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_6 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_6 | [] | [
"Definition:Subsequence",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Smallest Element",
"Well-Ordering Principle",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Mapping",
"Well-Ordering Principle",
"Definition:Well-Defined",
"Principle of Recursive Definition",
"Definition:Unique",
"Def... |
proofwiki-18652 | Sequence of P-adic Integers has Convergent Subsequence/Lemma 1 | there exists a $p$-adic digit $b_0$ such that:
:there exists infinitely many $n \in \N$ such that the canonical expansion of $x_n$ begins with the $p$-adic digit $b_0$ | === Case 1 ===
Let there exist $b \in \set{0, 1, \ldots , p - 2}$:
:there exists infinitely many $n \in \N$ such that the canonical expansion of $y_n$ begins with the $p$-adic digits $b$
Let $b_0 = b$ and the result holds.
{{qed|lemma}} | there exists a [[Definition:P-adic Digit|$p$-adic digit]] $b_0$ such that:
:there exists [[Definition:Infinite|infinitely many]] $n \in \N$ such that the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x_n$ begins with the [[Definition:P-adic Digit|$p$-adic digit]] $b_0$ | === Case 1 ===
Let there exist $b \in \set{0, 1, \ldots , p - 2}$:
:there exists [[Definition:Infinite|infinitely many]] $n \in \N$ such that the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $y_n$ begins with the [[Definition:P-adic Digit|$p$-adic digits]] $b$
Let $b_0 = b$ and the result holds.
... | Sequence of P-adic Integers has Convergent Subsequence/Lemma 1 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_1 | https://proofwiki.org/wiki/Sequence_of_P-adic_Integers_has_Convergent_Subsequence/Lemma_1 | [] | [
"Definition:P-adic Digit",
"Definition:Infinite",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] | [
"Definition:Infinite",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Digit"
] |
proofwiki-18653 | Real Power Function for Positive Integer Power is Continuous | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $f_n: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map {f_n} x = x^n$
Then $f_n$ is continuous on $\R$. | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\forall x \in \R: f_n$ is continuous on $\R$.
$\map P 0$ is the case:
:$\forall x \in \R: \map {f_0} x = x^0 = 1$
Thus it is seen that $f_0$ is the constant mapping.
It follows from Constant Real Function is Continuous tha... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $f_n: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map {f_n} x = x^n$
Then $f_n$ is [[Definition:Continuous Real Function|continuous]] on $\R$. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall x \in \R: f_n$ is [[Definition:Continuous Real Function|continuous]] on $\R$.
$\map P 0$ is the case:
:$\forall x \in \R: \map {f_0} x = x^0 = 1... | Real Power Function for Positive Integer Power is Continuous | https://proofwiki.org/wiki/Real_Power_Function_for_Positive_Integer_Power_is_Continuous | https://proofwiki.org/wiki/Real_Power_Function_for_Positive_Integer_Power_is_Continuous | [
"Continuous Real Functions",
"Integer Powers"
] | [
"Definition:Positive/Integer",
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Continuous Real Function",
"Definition:Constant Mapping",
"Constant Function is Continuous/Real Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
... |
proofwiki-18654 | Bound for Negative Part of Pointwise Sum of Functions | Let $X$ be a set.
Let $f, g : X \to \overline \R$ be extended real-valued function.
Suppose that the pointwise sum $f + g$ is well-defined, that is:
:there exists no $x \in X$ such that $\set {\map f x, \map g x} = \set {\infty, -\infty}$.
Then:
:$\paren {f + g}^- \le f^- + g^-$
where $\paren {f + g}^-$, $f^-$ and $... | Let $x \in X$.
From the definition of the negative part, we have:
:$\map {f^-} x = -\min \set {\map f x, 0}$
and:
:$\map {g^-} x = -\min \set {\map g x, 0}$
Suppose first that $\map f x$ and $\map g x$ are finite.
From Minimum Function in terms of Absolute Value, we then have:
:$\ds \map {f^-} x = \frac {\size {\map ... | Let $X$ be a [[Definition:Set|set]].
Let $f, g : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|extended real-valued function]].
Suppose that the [[Definition:Pointwise Addition|pointwise sum]] $f + g$ is well-defined, that is:
:there exists no $x \in X$ such that $\set {\map f x, \map g x} = \s... | Let $x \in X$.
From the definition of the [[Definition:Negative Part|negative part]], we have:
:$\map {f^-} x = -\min \set {\map f x, 0}$
and:
:$\map {g^-} x = -\min \set {\map g x, 0}$
Suppose first that $\map f x$ and $\map g x$ are finite.
From [[Minimum Function in terms of Absolute Value]], we then have:
... | Bound for Negative Part of Pointwise Sum of Functions | https://proofwiki.org/wiki/Bound_for_Negative_Part_of_Pointwise_Sum_of_Functions | https://proofwiki.org/wiki/Bound_for_Negative_Part_of_Pointwise_Sum_of_Functions | [
"Measure Theory",
"Negative Parts",
"Negative Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Pointwise Addition",
"Definition:Negative Part"
] | [
"Definition:Negative Part",
"Minimum Function in terms of Absolute Value",
"Minimum Function in terms of Absolute Value",
"Triangle Inequality",
"Category:Negative Parts"
] |
proofwiki-18655 | Real Polynomial Function is Differentiable | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $f_n: \R \to \R$ be a real polynomial function.
Then $f_n$ is differentiable over $\R$. | Let $f_n$ be an arbitrary real polynomial function of degree $n$.
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$f_n$ is differentiable over $\R$.
$\map P 0$ is the case $f_0$, where $f_0$ is of zero degree.
Such a real polynomial function is a constant function.
From ... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $f_n: \R \to \R$ be a [[Definition:Real Polynomial Function|real polynomial function]].
Then $f_n$ is [[Definition:Differentiable Real Function|differentiable]] over $\R$. | Let $f_n$ be an arbitrary [[Definition:Real Polynomial Function|real polynomial function]] of [[Definition:Degree of Polynomial|degree]] $n$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$f_n$ is [[D... | Real Polynomial Function is Differentiable | https://proofwiki.org/wiki/Real_Polynomial_Function_is_Differentiable | https://proofwiki.org/wiki/Real_Polynomial_Function_is_Differentiable | [
"Real Polynomial Functions",
"Differentiable Real Functions"
] | [
"Definition:Positive/Integer",
"Definition:Polynomial Function/Real",
"Definition:Differentiable Mapping/Real Function"
] | [
"Definition:Polynomial Function/Real",
"Definition:Degree of Polynomial",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Differentiable Mapping/Real Function",
"Definition:Degree of Polynomial/Zero",
"Definition:Polynomial Function/Real",
"Definition:Constant Mapping",
... |
proofwiki-18656 | Positive Part of Multiple of Function | Let $X$ be a set.
Let $f : X \to \overline \R$ be an extended real-valued function.
Let $\alpha$ be a real number.
Then:
:$\ds \paren {\alpha f}^+ = \begin{cases}\alpha f^+ & \alpha \ge 0 \\ -\alpha f^- & \alpha < 0\end{cases}$
where:
:$\paren {\alpha f}^+$ and $f^+$ are the positive parts of $\alpha f$ and $f$ respec... | Let $x \in X$.
First take $\alpha \ge 0$.
Suppose that $\map f x \ge 0$.
Then $\alpha \map f x \ge 0$.
So:
:$\max \set {\alpha \map f x, 0} = \alpha \map f x$
and:
:$\max \set {\map f x, 0} = \map f x$
So by the definition of the positive part and negative part, we have:
:$\map {\paren {\alpha f}^+} x = \alpha \map f ... | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real-valued function]].
Let $\alpha$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \paren {\alpha f}^+ = \begin{cases}\alpha f^+ & \alpha \ge 0 \\ -\alpha f^- & \alpha < 0\end{cas... | Let $x \in X$.
First take $\alpha \ge 0$.
Suppose that $\map f x \ge 0$.
Then $\alpha \map f x \ge 0$.
So:
:$\max \set {\alpha \map f x, 0} = \alpha \map f x$
and:
:$\max \set {\map f x, 0} = \map f x$
So by the definition of the [[Definition:Positive Part|positive part]] and [[Definition:Negative Part|negativ... | Positive Part of Multiple of Function | https://proofwiki.org/wiki/Positive_Part_of_Multiple_of_Function | https://proofwiki.org/wiki/Positive_Part_of_Multiple_of_Function | [
"Measure Theory",
"Positive Parts",
"Positive Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Real Number",
"Definition:Positive Part",
"Definition:Negative Part"
] | [
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Definition:Negative Part",
"Category:Positive Parts"
] |
proofwiki-18657 | Negative Part of Multiple of Function | Let $X$ be a set.
Let $f : X \to \overline \R$ be an extended real-valued function.
Let $\alpha$ be a real number.
Then:
:$\ds \paren {\alpha f}^- = \begin{cases}\alpha f^- & \alpha \ge 0 \\ -\alpha f^+ & \alpha < 0\end{cases}$
where:
:$\paren {\alpha f}^-$ and $f^-$ are the negative parts of $\alpha f$ and $f$ respec... | Let $x \in X$.
First take $\alpha \ge 0$.
Suppose that $\map f x \ge 0$.
Then $\alpha \map f x \ge 0$.
So:
:$-\min \set {\alpha \map f x, 0} = 0$
and:
:$-\min \set {\map f x, 0} = 0$
So by the definition of the positive part and negative part, we have:
:$\map {\paren {\alpha f}^-} x = 0$
and:
:$\map {f^-} x = 0$
So:
:... | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real-valued function]].
Let $\alpha$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \paren {\alpha f}^- = \begin{cases}\alpha f^- & \alpha \ge 0 \\ -\alpha f^+ & \alpha < 0\end{cas... | Let $x \in X$.
First take $\alpha \ge 0$.
Suppose that $\map f x \ge 0$.
Then $\alpha \map f x \ge 0$.
So:
:$-\min \set {\alpha \map f x, 0} = 0$
and:
:$-\min \set {\map f x, 0} = 0$
So by the definition of the [[Definition:Positive Part|positive part]] and [[Definition:Negative Part|negative part]], we have:
... | Negative Part of Multiple of Function | https://proofwiki.org/wiki/Negative_Part_of_Multiple_of_Function | https://proofwiki.org/wiki/Negative_Part_of_Multiple_of_Function | [
"Measure Theory",
"Negative Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Real Number",
"Definition:Negative Part",
"Definition:Positive Part"
] | [
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Positive Part",
"Definition:Negative Part",
"Category:Negative Parts"
] |
proofwiki-18658 | Pointwise Scalar Multiple of Measurable Function is Measurable | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f : X \to \overline \R$ be a $\Sigma$-measurable function.
Let $\alpha \in \overline \R$ be an extended real number.
Then the pointwise scalar multiple $\alpha f$ is $\Sigma$-measurable. | We want to show that:
:$\set {x \in X : \alpha \map f x \le t}$ is $\Sigma$-measurable for each $t \in \R$.
in each of the cases:
:$(1): \quad$ $\alpha = 0$
:$(2): \quad$ $0 < \alpha < \infty$
:$(3): \quad$ $-\infty < \alpha < 0$
:$(4): \quad$ $\alpha = \infty$
:$(5): \quad$ $\alpha = -\infty$
If $\alpha = 0$, then $... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f : X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Let $\alpha \in \overline \R$ be an [[Definition:Extended Real Number Line|extended real number]].
Then the [[Definition:Pointwise Scal... | We want to show that:
:$\set {x \in X : \alpha \map f x \le t}$ is [[Definition:Measurable Set|$\Sigma$-measurable]] for each $t \in \R$.
in each of the cases:
:$(1): \quad$ $\alpha = 0$
:$(2): \quad$ $0 < \alpha < \infty$
:$(3): \quad$ $-\infty < \alpha < 0$
:$(4): \quad$ $\alpha = \infty$
:$(5): \quad$ $\alpha =... | Pointwise Scalar Multiple of Measurable Function is Measurable | https://proofwiki.org/wiki/Pointwise_Scalar_Multiple_of_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Pointwise_Scalar_Multiple_of_Measurable_Function_is_Measurable | [
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Measurable Function",
"Definition:Extended Real Number Line",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Measurable Function"
] | [
"Definition:Measurable Set",
"Definition:Measurable Function",
"Constant Function is Measurable",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Characterization of Measurable Functions",
"Definition:Measurable Set",
"Definition:Extended Real Multiplication",... |
proofwiki-18659 | Characterization of Essentially Bounded Functions | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \R$ be a $\Sigma$-measurable function.
{{TFAE}}
:$(1) \quad$ $f$ is essentially bounded.
:$(2) \quad$ There exists a bounded function $g : X \to \R$ such that $f = g$ $\mu$-almost everywhere. | === $(1)$ implies $(2)$ ===
Suppose that there exists $c \in \R$ such that:
:$\map \mu {\set {x \in X : \size {\map f x} > c} } = 0$
Let:
:$A = \set {x \in X : \size {\map f x} \le c}$
Define a function $g : X \to \overline \R$ by:
:$\map g x = \begin{cases}\map f x & x \in A \\ 0 & x \not \in A\end{cases}$
Then:
:$\... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
{{TFAE}}
:$(1) \quad$ $f$ is [[Definition:Essentially Bounded Function|essentially bounded]].
:$(2) \quad$ There exists a [[Definition:Bounded Rea... | === $(1)$ implies $(2)$ ===
Suppose that there exists $c \in \R$ such that:
:$\map \mu {\set {x \in X : \size {\map f x} > c} } = 0$
Let:
:$A = \set {x \in X : \size {\map f x} \le c}$
Define a [[Definition:Extended Real-Valued Function|function]] $g : X \to \overline \R$ by:
:$\map g x = \begin{cases}\map f x ... | Characterization of Essentially Bounded Functions | https://proofwiki.org/wiki/Characterization_of_Essentially_Bounded_Functions | https://proofwiki.org/wiki/Characterization_of_Essentially_Bounded_Functions | [
"Measure Theory"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Essentially Bounded Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Almost Everywhere"
] | [
"Definition:Extended Real-Valued Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Almost Everywhere",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Almost Everywhere"
] |
proofwiki-18660 | Triangle Inequality for Integrals/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.
Then $\cmod f$ is $\mu$-integrable and:
:$\ds \cmod {\int f \rd \mu} \le \int \cmod f \rd \mu$ | Let $\struct {\R, \map \BB \R}$ be the real numbers made into a measurable space with its Borel $\sigma$-algebra.
From Complex Modulus of Measurable Function is Measurable, $\cmod f$ is $\Sigma/\map \BB \R$-measurable.
We have:
{{begin-eqn}}
{{eqn | l = \size {\map \Re f}^2 + \size {\map \Im f}^2
| o = \le
| r = \... | 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 ... | Let $\struct {\R, \map \BB \R}$ be the [[Definition:Real Number|real numbers]] made into a [[Definition:Measurable Space|measurable space]] with its [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]].
From [[Complex Modulus of Measurable Function is Measurable]], $\cmod f$ is [[Definition:Measurable Mapping|$\S... | Triangle Inequality for Integrals/Complex Function | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Complex_Function | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Complex_Function | [
"Triangle Inequality for Integrals"
] | [
"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"
] | [
"Definition:Real Number",
"Definition:Measurable Space",
"Definition:Borel Sigma-Algebra",
"Complex Modulus of Measurable Function is Measurable",
"Definition:Measurable Mapping",
"Definition:Square Root",
"Definition:Integrable Function/Measure Space/Complex Function",
"Definition:Integrable Function... |
proofwiki-18661 | Dimension of Double Dual | Let $G^{**}$ be the double dual of $G$.
Then $G^{**}$ is also $n$-dimensional. | By definition, the double dual of $G$ is the algebraic dual of the algebraic dual $G^*$ of $G$.
From Dimension of Algebraic Dual:
:$\map \dim {G^**} = \map \dim {G^*}$
Also from Dimension of Algebraic Dual::
:$\map \dim {G^*} = \map \dim G$
Hence the result.
{{Qed}} | Let $G^{**}$ be the [[Definition:Double Dual|double dual]] of $G$.
Then $G^{**}$ is also [[Definition:Dimension (Linear Algebra)|$n$-dimensional]]. | By definition, the [[Definition:Double Dual|double dual]] of $G$ is the [[Definition:Algebraic Dual|algebraic dual]] of the [[Definition:Algebraic Dual|algebraic dual]] $G^*$ of $G$.
From [[Dimension of Algebraic Dual]]:
:$\map \dim {G^**} = \map \dim {G^*}$
Also from [[Dimension of Algebraic Dual]]::
:$\map \dim {G^... | Dimension of Double Dual | https://proofwiki.org/wiki/Dimension_of_Double_Dual | https://proofwiki.org/wiki/Dimension_of_Double_Dual | [
"Algebraic Duals"
] | [
"Definition:Algebraic Dual/Double Dual",
"Definition:Dimension (Linear Algebra)"
] | [
"Definition:Algebraic Dual/Double Dual",
"Definition:Algebraic Dual",
"Definition:Algebraic Dual",
"Dimension of Algebraic Dual",
"Dimension of Algebraic Dual"
] |
proofwiki-18662 | Expression for Set of Points at which Sequence of Functions does not Converge to Given Function | Let $X$ be a set.
Let $f : X \to \R$ be a real function.
For each $n \in \N$, let $f_n : X \to \R$ be a real function.
Then we have:
:$\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N... | It helps to rewrite:
{{begin-eqn}}
{{eqn | l = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \in X : \size {\map {f_n} x - \map f x} \ge \frac 1 k}
| r = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \set {x \in X : \size {\map {f_n} x - \map f x} ... | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \R$ be a [[Definition:Real Function|real function]].
For each $n \in \N$, let $f_n : X \to \R$ be a [[Definition:Real Function|real function]].
Then we have:
:$\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \b... | It helps to rewrite:
{{begin-eqn}}
{{eqn | l = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \in X : \size {\map {f_n} x - \map f x} \ge \frac 1 k}
| r = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \set {x \in X : \size {\map {f_n} x - \map f x}... | Expression for Set of Points at which Sequence of Functions does not Converge to Given Function | https://proofwiki.org/wiki/Expression_for_Set_of_Points_at_which_Sequence_of_Functions_does_not_Converge_to_Given_Function | https://proofwiki.org/wiki/Expression_for_Set_of_Points_at_which_Sequence_of_Functions_does_not_Converge_to_Given_Function | [
"Convergence"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Real Function"
] | [
"Definition:Convergent Sequence/Real Numbers",
"Definition:Real Number",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Real Number",
"Definition:Set Equality",
"Category:Convergence"
] |
proofwiki-18663 | Set of Points at which Sequence of Measurable Functions does not Converge to Given Measurable Function is Measurable | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \R$ be a $\Sigma$-measurable function.
For each $n \in \N$, let $f_n : X \to \R$ be a $\Sigma$-measurable function.
Then:
:$\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x}$ is $\Sigma$-measurable. | From Expression for Set of Points at which Sequence of Functions does not Converge to Given Function, we have:
:$\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \in X ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
For each $n \in \N$, let $f_n : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Then:
:$\ds \set {x \in X : \sequ... | From [[Expression for Set of Points at which Sequence of Functions does not Converge to Given Function]], we have:
:$\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \... | Set of Points at which Sequence of Measurable Functions does not Converge to Given Measurable Function is Measurable | https://proofwiki.org/wiki/Set_of_Points_at_which_Sequence_of_Measurable_Functions_does_not_Converge_to_Given_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Set_of_Points_at_which_Sequence_of_Measurable_Functions_does_not_Converge_to_Given_Measurable_Function_is_Measurable | [
"Measurable Functions",
"Convergence"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Set"
] | [
"Expression for Set of Points at which Sequence of Functions does not Converge to Given Function",
"Pointwise Difference of Measurable Functions is Measurable",
"Definition:Measurable Function",
"Absolute Value of Measurable Function is Measurable",
"Definition:Measurable Function",
"Characterization of M... |
proofwiki-18664 | Pointwise Convergence implies Convergence Almost Everywhere | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \R$ be a $\Sigma$-measurable function.
For each $n \mathop \in \N$, let $f_n : X \to \R$ be a $\Sigma$-measurable function.
Then:
:if $\sequence {f_n}_{n \mathop \in \N}$ converges pointwise to $f$, it converges almost everywhere to $f$. | If $\sequence {f_n}_{n \mathop \in \N}$ converges pointwise to $f$, then:
:$\sequence {\map {f_n} x}_{n \mathop \in \N}$ converges to $\map f x$ for each $x \in X$.
So:
:$\set {x \in X : \sequence {\map {f_n} x}_{n \mathop \in \N} \text { does not converge to } \map f x} = \O$
From Measure of Empty Set is Zero, we ha... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
For each $n \mathop \in \N$, let $f_n : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Then:
:if $\sequence {f_n... | If $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Pointwise Convergence|converges pointwise]] to $f$, then:
:$\sequence {\map {f_n} x}_{n \mathop \in \N}$ [[Definition:Convergent Sequence|converges]] to $\map f x$ for each $x \in X$.
So:
:$\set {x \in X : \sequence {\map {f_n} x}_{n \mathop \in \N} \text { does... | Pointwise Convergence implies Convergence Almost Everywhere | https://proofwiki.org/wiki/Pointwise_Convergence_implies_Convergence_Almost_Everywhere | https://proofwiki.org/wiki/Pointwise_Convergence_implies_Convergence_Almost_Everywhere | [
"Measure Theory",
"Convergence Almost Everywhere",
"Convergence Almost Everywhere"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Pointwise Convergence",
"Definition:Convergence Almost Everywhere"
] | [
"Definition:Pointwise Convergence",
"Definition:Convergent Sequence",
"Measure of Empty Set is Zero",
"Definition:Convergence Almost Everywhere",
"Category:Convergence Almost Everywhere"
] |
proofwiki-18665 | Convergence in Mean implies Convergence in Measure | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \R$ be a $\Sigma$-measurable function.
For each $n \mathop \in \N$, let $f_n : X \to \R$ be a $\Sigma$-measurable function.
Then:
:if $\sequence {f_n}_{n \mathop \in \N}$ converges in mean to $f$, it converges in measure to $f$. | From Pointwise Difference of Measurable Functions is Measurable:
:$f_n - f$ is $\Sigma$-measurable for each $n \in \N$.
Let $\epsilon > 0$ be a real number.
From Markov's Inequality, we then have:
:$\ds \map \mu {\set {x \in X : \size {\map {f_n} x - \map f x} > \epsilon} } \le \frac 1 \epsilon \int \size {f_n - f} \... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
For each $n \mathop \in \N$, let $f_n : X \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Then:
:if $\sequence {f_n... | From [[Pointwise Difference of Measurable Functions is Measurable]]:
:$f_n - f$ is [[Definition:Measurable Function|$\Sigma$-measurable]] for each $n \in \N$.
Let $\epsilon > 0$ be a [[Definition:Real Number|real number]].
From [[Markov's Inequality]], we then have:
:$\ds \map \mu {\set {x \in X : \size {\map {f_... | Convergence in Mean implies Convergence in Measure | https://proofwiki.org/wiki/Convergence_in_Mean_implies_Convergence_in_Measure | https://proofwiki.org/wiki/Convergence_in_Mean_implies_Convergence_in_Measure | [
"Convergence in Measure",
"Measure Theory",
"Convergence in Mean",
"Convergence in Mean",
"Convergence in Measure"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Convergence in Mean",
"Definition:Convergence in Measure"
] | [
"Pointwise Difference of Measurable Functions is Measurable",
"Definition:Measurable Function",
"Definition:Real Number",
"Markov's Inequality",
"Definition:Convergence in Mean",
"Squeeze Theorem",
"Definition:Convergence in Measure",
"Category:Convergence in Mean",
"Category:Convergence in Measure"... |
proofwiki-18666 | Basis for R-Module R | Let $\struct {R, +, \times}$ be a ring with unity whose unity is $1_R$.
Let $\struct {R, +_R, \circ}_R$ denote the $R$-module $R$.
Then $\set {1_R}$ is a basis for $\struct {R, +_R, \circ}_R$. | From Dimension of $R$-Module $R$ is $1$ we have that $\struct {R, +_R, \circ}_R$ is $1$-dimensional.
From Standard Ordered Basis is Basis it follows directly that $\set {1_R}$ is a basis for $\struct {R, +_R, \circ}_R$.
{{qed}} | Let $\struct {R, +, \times}$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\struct {R, +_R, \circ}_R$ denote the [[Definition:R-Module R|$R$-module $R$]].
Then $\set {1_R}$ is a [[Definition:Basis (Linear Algebra)|basis]] for $\struct {R, +_R, \circ}_R$. | From [[Dimension of R-Module R is 1|Dimension of $R$-Module $R$ is $1$]] we have that $\struct {R, +_R, \circ}_R$ is [[Definition:Dimension (Linear Algebra)|$1$-dimensional]].
From [[Standard Ordered Basis is Basis]] it follows directly that $\set {1_R}$ is a [[Definition:Basis (Linear Algebra)|basis]] for $\struct {R... | Basis for R-Module R | https://proofwiki.org/wiki/Basis_for_R-Module_R | https://proofwiki.org/wiki/Basis_for_R-Module_R | [
"Module on Cartesian Product"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Module on Cartesian Product/Special Case",
"Definition:Basis (Linear Algebra)"
] | [
"Dimension of R-Module R is 1",
"Definition:Dimension (Linear Algebra)",
"Standard Ordered Basis is Basis",
"Definition:Basis (Linear Algebra)"
] |
proofwiki-18667 | Distribution acting on Sequence of Test Functions without common Support is not Continuous | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \DD \R$ be a sequence of test functions.
Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ does not have the common compact support.
Then $T$ acting on $\sequence {\phi_n}_{n \mathop \in \N}$ is not continuous. | Let $\operatorname {III} \in \map {\DD'} \R$ be the Dirac comb.
Let $\mathbf 0 : \R \to 0$ be the zero mapping.
Let $\phi \in \map \DD \R$ be a test function with compact support $K = \closedint 0 1$ such that:
:$\forall x \in K : \map \phi x > 0$
Let $\phi_n \in \map \DD \R$ be a test function sequence such that:
:$\d... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \DD \R$ be a [[Definition:Sequence|sequence]] of [[Definition:Test Function|test functions]].
Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ does not have the common [[Defin... | Let $\operatorname {III} \in \map {\DD'} \R$ be the [[Definition:Dirac Comb|Dirac comb]].
Let $\mathbf 0 : \R \to 0$ be the [[Definition:Zero Mapping|zero mapping]].
Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]] with [[Definition:Compact Subset of Real Euclidean Space|compact]] [[Definiti... | Distribution acting on Sequence of Test Functions without common Support is not Continuous | https://proofwiki.org/wiki/Distribution_acting_on_Sequence_of_Test_Functions_without_common_Support_is_not_Continuous | https://proofwiki.org/wiki/Distribution_acting_on_Sequence_of_Test_Functions_without_common_Support_is_not_Continuous | [
"Convergence",
"Continuity"
] | [
"Definition:Schwartz Distribution",
"Definition:Sequence",
"Definition:Test Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Continuous Mapping",
"Definition:Continuous Mapping/Distribution"
] | [
"Definition:Sampling Function",
"Definition:Zero Mapping",
"Definition:Test Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Test Function",
"Definition:Sequence",
"Definition:Test Function",
"Definition:Support of Continuous ... |
proofwiki-18668 | Basis for Set of Linear Transformations | Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module such that $\map \dim G = n$.
Let $\struct {H, +_H, \circ}_R$ be a unitary $R$-module such that $\map \dim H = m$.
Let $\map {\LL_R} {G, H}$ be the set of all linear transformati... | Let $B = \set {\phi_{i j}: i \in \closedint 1 n, j \in \closedint 1 m}$.
Let $\ds \sum_{j \mathop = 1}^m \sum_{i \mathop = 1}^n \lambda_{i j} \phi_{i j} = 0$.
Then:
:$\ds \forall k \in \closedint 1 n: 0 = \sum_{j \mathop = 1}^m \sum_{i \mathop = 1}^n \lambda_{i j} \map {\phi_{i j} } {a_k} = \sum_{j \mathop = 1}^m \lamb... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\struct {G, +_G, \circ}_R$ be a [[Definition:Unitary Module|unitary $R$-module]] such that $\map \dim G = n$.
Let $\struct {H, +... | Let $B = \set {\phi_{i j}: i \in \closedint 1 n, j \in \closedint 1 m}$.
Let $\ds \sum_{j \mathop = 1}^m \sum_{i \mathop = 1}^n \lambda_{i j} \phi_{i j} = 0$.
Then:
:$\ds \forall k \in \closedint 1 n: 0 = \sum_{j \mathop = 1}^m \sum_{i \mathop = 1}^n \lambda_{i j} \map {\phi_{i j} } {a_k} = \sum_{j \mathop = 1}^m \la... | Basis for Set of Linear Transformations | https://proofwiki.org/wiki/Basis_for_Set_of_Linear_Transformations | https://proofwiki.org/wiki/Basis_for_Set_of_Linear_Transformations | [
"Linear Transformations"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unitary Module over Ring",
"Definition:Unitary Module over Ring",
"Definition:Set of All Linear Transformations",
"Definition:Ordered Basis",
"Definition:Ordered Basis",
"Defini... | [
"Definition:Linearly Independent/Set",
"Definition:Sequence",
"Definition:Scalar/Module",
"Linear Transformation of Generated Module",
"Definition:Generator of Module"
] |
proofwiki-18669 | Dimension of R-Module R is 1 | Let $\struct {R, +, \times}$ be a ring whose unity is $1_R$.
Let $\struct {R, +_R, \circ}_R$ denote the $R$-module $R$.
Then the dimension of $\struct {R, +_R, \circ}_R$ is $1$. | {{improve|This proof applies only for a ring with unity. Needs to be expanded to any ring.}}
We have by definition that the $R$-module $R$ is the special case of the $R$-module $R^n$ where $n = 1$.
From $R$-module $R^n$ is $n$-Dimensional it follows that $\struct {R, +_R, \circ}_R$ is $1$-dimensional.
{{qed}}
Category:... | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\struct {R, +_R, \circ}_R$ denote the [[Definition:R-Module R|$R$-module $R$]].
Then the [[Definition:Dimension (Linear Algebra)|dimension]] of $\struct {R, +_R, \circ}_R$ is $1$. | {{improve|This proof applies only for a [[Definition:Ring with Unity|ring with unity]]. Needs to be expanded to any ring.}}
We have by definition that the [[Definition:R-Module R|$R$-module $R$]] is the special case of the [[Definition:Module on Cartesian Product|$R$-module $R^n$]] where $n = 1$.
From [[R-Module R^n ... | Dimension of R-Module R is 1 | https://proofwiki.org/wiki/Dimension_of_R-Module_R_is_1 | https://proofwiki.org/wiki/Dimension_of_R-Module_R_is_1 | [
"Module on Cartesian Product"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Module on Cartesian Product/Special Case",
"Definition:Dimension (Linear Algebra)"
] | [
"Definition:Ring with Unity",
"Definition:Module on Cartesian Product/Special Case",
"Definition:Module on Cartesian Product",
"R-Module R^n is n-Dimensional",
"Definition:Dimension (Linear Algebra)",
"Category:Module on Cartesian Product"
] |
proofwiki-18670 | Underlying Mapping of Evaluation Linear Transformation is Element of Double Dual | Let $\struct {R, +, \times}$ be a commutative ring with unity.
Let $G$ be an $R$-module.
Let $G^*$ be the algebraic dual of $G$.
Let $G^{**}$ be the double dual of $G$.
For each $x \in G$, let $x^\wedge: G^* \to R$ be defined as:
:$\forall t \in G^*: \map {x^\wedge} t = \map t x$
Then:
:$x^\wedge \in G^{**}$ | We have that $x^\wedge$ is a mapping from $G^* \to R$.
It remains to be demonstrates that $x^\wedge$ is in fact a linear transformation.
Hence we need to show that:
:$(1): \quad \forall u, v \in G^*: \map {x^\wedge} {u + v} = \map {x^\wedge} u + \map {x^\wedge} v$
:$(2): \quad \forall u \in G^*: \forall \lambda \in R: ... | Let $\struct {R, +, \times}$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $G$ be an [[Definition:Module over Ring|$R$-module]].
Let $G^*$ be the [[Definition:Algebraic Dual|algebraic dual]] of $G$.
Let $G^{**}$ be the [[Definition:Double Dual|double dual]] of $G$.
For each $x \i... | We have that $x^\wedge$ is a [[Definition:Mapping|mapping]] from $G^* \to R$.
It remains to be demonstrates that $x^\wedge$ is in fact a [[Definition:Linear Transformation|linear transformation]].
Hence we need to show that:
:$(1): \quad \forall u, v \in G^*: \map {x^\wedge} {u + v} = \map {x^\wedge} u + \map {x^\w... | Underlying Mapping of Evaluation Linear Transformation is Element of Double Dual | https://proofwiki.org/wiki/Underlying_Mapping_of_Evaluation_Linear_Transformation_is_Element_of_Double_Dual | https://proofwiki.org/wiki/Underlying_Mapping_of_Evaluation_Linear_Transformation_is_Element_of_Double_Dual | [
"Linear Transformations",
"Evaluation Linear Transformations (Module Theory)"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Module over Ring",
"Definition:Algebraic Dual",
"Definition:Algebraic Dual/Double Dual"
] | [
"Definition:Mapping",
"Definition:Linear Transformation"
] |
proofwiki-18671 | Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings/Unitary | Let $\struct {H, +_H, \circ}_R$ be a unitary module.
Then $\map {\LL_R} {G, H}$ is also a unitary module. | From Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings, $\map {\LL_R} {G, H}$ is a module.
It remains to be shown that $\map {\LL_R} {G, H}$ is a unitary module, that is:
:$\forall \phi \in \map {\LL_R} {G, H}: 1_R \circ \phi = \phi$
So, let $\struct {H, +_H, \circ}_R$ be a u... | Let $\struct {H, +_H, \circ}_R$ be a [[Definition:Unitary Module over Ring|unitary module]].
Then $\map {\LL_R} {G, H}$ is also a [[Definition:Unitary Module over Ring|unitary module]]. | From [[Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings]], $\map {\LL_R} {G, H}$ is a [[Definition:Module over Ring|module]].
It remains to be shown that $\map {\LL_R} {G, H}$ is a [[Definition:Unitary Module over Ring|unitary module]], that is:
:$\forall \phi \in \map {\L... | Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings/Unitary | https://proofwiki.org/wiki/Set_of_Linear_Transformations_over_Commutative_Ring_forms_Submodule_of_Module_of_All_Mappings/Unitary | https://proofwiki.org/wiki/Set_of_Linear_Transformations_over_Commutative_Ring_forms_Submodule_of_Module_of_All_Mappings/Unitary | [
"Linear Transformations",
"Unitary Modules"
] | [
"Definition:Unitary Module over Ring",
"Definition:Unitary Module over Ring"
] | [
"Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings",
"Definition:Module over Ring",
"Definition:Unitary Module over Ring",
"Definition:Unitary Module over Ring"
] |
proofwiki-18672 | Annihilator is Submodule of Algebraic Dual/Corollary | Let $N$ be a submodule of $G^*$.
Let $G^{**}$ be the algebraic dual of $G^*$.
Then the annihilator $N^\circ$ of $N$ is a submodule of $G^{**}$. | Follows directly as an example of Annihilator is Submodule of Algebraic Dual.
{{qed}} | Let $N$ be a [[Definition:Submodule|submodule]] of $G^*$.
Let $G^{**}$ be the [[Definition:Algebraic Dual|algebraic dual]] of $G^*$.
Then the [[Definition:Annihilator on Algebraic Dual|annihilator]] $N^\circ$ of $N$ is a [[Definition:Submodule|submodule]] of $G^{**}$. | Follows directly as an example of [[Annihilator is Submodule of Algebraic Dual]].
{{qed}} | Annihilator is Submodule of Algebraic Dual/Corollary | https://proofwiki.org/wiki/Annihilator_is_Submodule_of_Algebraic_Dual/Corollary | https://proofwiki.org/wiki/Annihilator_is_Submodule_of_Algebraic_Dual/Corollary | [
"Annihilator is Submodule of Algebraic Dual"
] | [
"Definition:Submodule",
"Definition:Algebraic Dual",
"Definition:Annihilator on Algebraic Dual",
"Definition:Submodule"
] | [
"Annihilator is Submodule of Algebraic Dual"
] |
proofwiki-18673 | Dimension of Annihilator on Algebraic Dual | :$M^\circ$ is an $\paren {n - m}$-dimensional subspace of $G^*$. | Let $\sequence {a_n}$ be an ordered basis of $G$ such that $\sequence {a_m}$ is an ordered basis of $M$.
Let $\sequence { {a_n}'}$ be the ordered dual basis of $G^*$.
Let $\ds t = \sum_{k \mathop = 1}^n \lambda_k {a_k}' \in M^\circ$.
Then:
{{begin-eqn}}
{{eqn | q = \forall j \in \closedint 1 m
| l = \lambda_j
... | :$M^\circ$ is an [[Definition:Dimension of Vector Space|$\paren {n - m}$-dimensional]] [[Definition:Vector Subspace|subspace]] of $G^*$. | Let $\sequence {a_n}$ be an [[Definition:Ordered Basis|ordered basis]] of $G$ such that $\sequence {a_m}$ is an ordered basis of $M$.
Let $\sequence { {a_n}'}$ be the [[Definition:Ordered Dual Basis|ordered dual basis]] of $G^*$.
Let $\ds t = \sum_{k \mathop = 1}^n \lambda_k {a_k}' \in M^\circ$.
Then:
{{begin-eqn}}... | Dimension of Annihilator on Algebraic Dual | https://proofwiki.org/wiki/Dimension_of_Annihilator_on_Algebraic_Dual | https://proofwiki.org/wiki/Dimension_of_Annihilator_on_Algebraic_Dual | [
"Annihilators",
"Vector Subspaces"
] | [
"Definition:Dimension of Vector Space",
"Definition:Vector Subspace"
] | [
"Definition:Ordered Basis",
"Definition:Ordered Dual Basis",
"Definition:Linear Combination"
] |
proofwiki-18674 | Annihilator of Annihilator on Algebraic Dual of Subspace is Image under Evaluation Isomorphism | :$M^{\circ \circ} = J \sqbrk M$
where $J \sqbrk M$ denotes the image of $M$ under $J$. | Let Dimension of Annihilator on Algebraic Dual be applied to $M^\circ$ instead of $M$.
It is seen that the annihilator $M^{\circ \circ}$ of $M^\circ$ has dimension $n - \paren {n - m} = m$.
But clearly:
:$J \sqbrk M \subseteq M^{\circ \circ}$.
As $J$ is an isomorphism, $J \sqbrk M$ has dimension $m$.
So by Dimension of... | :$M^{\circ \circ} = J \sqbrk M$
where $J \sqbrk M$ denotes the [[Definition:Image of Subset under Mapping|image of $M$ under $J$]]. | Let [[Dimension of Annihilator on Algebraic Dual]] be applied to $M^\circ$ instead of $M$.
It is seen that the [[Definition:Annihilator on Algebraic Dual|annihilator]] $M^{\circ \circ}$ of $M^\circ$ has [[Definition:Dimension of Vector Space|dimension]] $n - \paren {n - m} = m$.
But clearly:
:$J \sqbrk M \subseteq M^... | Annihilator of Annihilator on Algebraic Dual of Subspace is Image under Evaluation Isomorphism | https://proofwiki.org/wiki/Annihilator_of_Annihilator_on_Algebraic_Dual_of_Subspace_is_Image_under_Evaluation_Isomorphism | https://proofwiki.org/wiki/Annihilator_of_Annihilator_on_Algebraic_Dual_of_Subspace_is_Image_under_Evaluation_Isomorphism | [
"Annihilators",
"Vector Subspaces"
] | [
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Dimension of Annihilator on Algebraic Dual",
"Definition:Annihilator on Algebraic Dual",
"Definition:Dimension of Vector Space",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Definition:Dimension of Vector Space",
"Dimension of Proper Subspace is... |
proofwiki-18675 | Mapping to Annihilator on Algebraic Dual is Bijection | Let $G_m$ denote the set of all $m$-dimensional subspaces of $G$.
Let ${G^*}_{n - m}$ denote the set of all $n - m$-dimensional subspaces of $G^*$.
Let $\phi: G_m \to {G^*}_{n - m}$ be the mapping from $G_m$ to the power set of ${G^*}_{n - m}$ defined as:
:$\forall M \in \powerset G: \map \phi M = M^\circ$
Then $\phi$ ... | From Annihilator of Annihilator on Algebraic Dual of Subspace is Image under Evaluation Isomorphism, we have that:
:$M^{\circ \circ} = J \sqbrk M$
From Evaluation Isomorphism is Isomorphism, $J: G \to G^{**}$ is a bijection.
{{finish|I may return to this if I can view it once more with a clear head. It's not completely... | Let $G_m$ denote the [[Definition:Set|set]] of all [[Definition:Dimension of Vector Space|$m$-dimensional]] [[Definition:Vector Subspace|subspaces]] of $G$.
Let ${G^*}_{n - m}$ denote the [[Definition:Set|set]] of all [[Definition:Dimension of Vector Space|$n - m$-dimensional]] [[Definition:Vector Subspace|subspaces]]... | From [[Annihilator of Annihilator on Algebraic Dual of Subspace is Image under Evaluation Isomorphism]], we have that:
:$M^{\circ \circ} = J \sqbrk M$
From [[Evaluation Isomorphism is Isomorphism]], $J: G \to G^{**}$ is a [[Definition:Bijection|bijection]].
{{finish|I may return to this if I can view it once more wit... | Mapping to Annihilator on Algebraic Dual is Bijection | https://proofwiki.org/wiki/Mapping_to_Annihilator_on_Algebraic_Dual_is_Bijection | https://proofwiki.org/wiki/Mapping_to_Annihilator_on_Algebraic_Dual_is_Bijection | [
"Annihilators",
"Vector Subspaces"
] | [
"Definition:Set",
"Definition:Dimension of Vector Space",
"Definition:Vector Subspace",
"Definition:Set",
"Definition:Dimension of Vector Space",
"Definition:Vector Subspace",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Bijection"
] | [
"Annihilator of Annihilator on Algebraic Dual of Subspace is Image under Evaluation Isomorphism",
"Evaluation Isomorphism is Isomorphism",
"Definition:Bijection"
] |
proofwiki-18676 | Inverse of Mapping to Annihilator on Algebraic Dual is Bijection | The inverse of $\phi$ is the bijection:
:$N \to \map {J^\gets} {N^\circ}$ | By definition::
:$\paren {\map {J^\gets} {N^\circ} }^\circ = \set {z \in G^*: \forall x \in G: \forall t \in N: \map t x = 0: \map z x = 0}$
Thus:
:$N \subseteq \paren {\map {J^\gets} {N^\circ} }^\circ$
But as $\paren {\map {J^\gets} {N^\circ} }^\circ$ has dimension $n - \paren {n - p} = p$, it follows that $N = \paren... | The [[Definition:Inverse Mapping|inverse]] of $\phi$ is the [[Definition:Bijection|bijection]]:
:$N \to \map {J^\gets} {N^\circ}$ | By definition::
:$\paren {\map {J^\gets} {N^\circ} }^\circ = \set {z \in G^*: \forall x \in G: \forall t \in N: \map t x = 0: \map z x = 0}$
Thus:
:$N \subseteq \paren {\map {J^\gets} {N^\circ} }^\circ$
But as $\paren {\map {J^\gets} {N^\circ} }^\circ$ has dimension $n - \paren {n - p} = p$, it follows that $N = \par... | Inverse of Mapping to Annihilator on Algebraic Dual is Bijection | https://proofwiki.org/wiki/Inverse_of_Mapping_to_Annihilator_on_Algebraic_Dual_is_Bijection | https://proofwiki.org/wiki/Inverse_of_Mapping_to_Annihilator_on_Algebraic_Dual_is_Bijection | [
"Annihilators",
"Vector Subspaces"
] | [
"Definition:Inverse Mapping",
"Definition:Bijection"
] | [
"Dimension of Proper Subspace is Less Than its Superspace",
"Definition:Inverse Mapping/Definition 2"
] |
proofwiki-18677 | Existence of Scalar for Vector Subspace Dimension One Less | Let $\sequence {\beta_n}$ be a sequence of scalars such that:
:$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \beta_1 \lambda_1 + \cdots + \beta_n \lambda_n = 0}$
Then there is a non-zero scalar $\gamma$ such that:
:$\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$ | Let $\sequence { {e_n}'}$ be the ordered basis of $\paren {K^n}^*$ dual to the standard ordered basis of $K^n$.
Let $M = \map \ker \psi$, where $\ds \psi = \sum_{k \mathop = 1}^n \beta_k {e_k}'$.
From Equivalent Statements for Vector Subspace Dimension One Less:
:$\psi = M^\circ$
As $M^\circ$ is one-dimensional and sin... | Let $\sequence {\beta_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Scalar (Vector Space)|scalars]] such that:
:$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \beta_1 \lambda_1 + \cdots + \beta_n \lambda_n = 0}$
Then there is a non-zero [[Definition:Scalar (Vector Space)|scalar]] $\gamma$ such ... | Let $\sequence { {e_n}'}$ be the [[Definition:Ordered Basis|ordered basis]] of $\paren {K^n}^*$ [[Definition:Ordered Dual Basis|dual to]] the [[Definition:Standard Ordered Basis|standard ordered basis]] of $K^n$.
Let $M = \map \ker \psi$, where $\ds \psi = \sum_{k \mathop = 1}^n \beta_k {e_k}'$.
From [[Equivalent Sta... | Existence of Scalar for Vector Subspace Dimension One Less | https://proofwiki.org/wiki/Existence_of_Scalar_for_Vector_Subspace_Dimension_One_Less | https://proofwiki.org/wiki/Existence_of_Scalar_for_Vector_Subspace_Dimension_One_Less | [
"Linear Algebra"
] | [
"Definition:Sequence",
"Definition:Scalar/Vector Space",
"Definition:Scalar/Vector Space"
] | [
"Definition:Ordered Basis",
"Definition:Ordered Dual Basis",
"Definition:Standard Ordered Basis",
"Equivalent Statements for Vector Subspace Dimension One Less",
"Definition:Dimension of Vector Space"
] |
proofwiki-18678 | Kernel of Transpose of Linear Transformation is Annihilator of Image | Let $G$ and $H$ be $n$-dimensional vector spaces over a field.
Let $\map \LL {G, H}$ be the set of all linear transformations from $G$ to $H$.
Let $u \in \map \LL {G, H}$.
Let $u^\intercal$ be the transpose of $u$.
Then:
:$\map \ker {u^\intercal}$ is the annihilator of the image of $u$
where $\map \ker {u^\intercal}$ d... | From the definitions of:
:the transpose $u^\intercal$
:the annihilator $\paren {\map u G}^\circ$
it follows that:
:$\map {u^\intercal} y = 0 \iff y \in \paren {\map u G}^\circ$
Thus:
:$\map \ker {u^\intercal} = \paren {\map u G}^\circ$
{{qed}} | Let $G$ and $H$ be [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]].
Let $\map \LL {G, H}$ be [[Definition:Set of All Linear Transformations|the set of all linear transformations]] from $G$ to $H$.
Let $u \in \map \... | From the definitions of:
:the [[Definition:Transpose of Linear Transformation|transpose]] $u^\intercal$
:the [[Definition:Annihilator|annihilator]] $\paren {\map u G}^\circ$
it follows that:
:$\map {u^\intercal} y = 0 \iff y \in \paren {\map u G}^\circ$
Thus:
:$\map \ker {u^\intercal} = \paren {\map u G}^\circ$
{{qed... | Kernel of Transpose of Linear Transformation is Annihilator of Image | https://proofwiki.org/wiki/Kernel_of_Transpose_of_Linear_Transformation_is_Annihilator_of_Image | https://proofwiki.org/wiki/Kernel_of_Transpose_of_Linear_Transformation_is_Annihilator_of_Image | [
"Linear Algebra"
] | [
"Definition:Dimension of Vector Space",
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Set of All Linear Transformations",
"Definition:Transpose of Linear Transformation",
"Definition:Annihilator on Algebraic Dual",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Defin... | [
"Definition:Transpose of Linear Transformation",
"Definition:Annihilator"
] |
proofwiki-18679 | Image of Transpose of Linear Transformation is Annihilator of Kernel | Let $G$ and $H$ be $n$-dimensional vector spaces over a field.
Let $\map \LL {G, H}$ be the set of all linear transformations from $G$ to $H$.
Let $u \in \map \LL {G, H}$.
Let $u^\intercal$ be the transpose of $u$.
Then:
:The image of $u^\intercal$ is the annihilator of $\map \ker u$.
where $\map \ker u$ denotes the ke... | Let $x \in \map \ker u$.
Let $H^*$ be the algebraic dual of $H$.
Let $\innerprod x t$ be the evaluation linear transformation.
Then:
:$\forall y \in H^*: \innerprod x {\map {u^\intercal} y} = \innerprod {\map u x} y = \innerprod 0 y = 0$
So:
:$\map {u^\intercal} {H^*} \subseteq \paren {\map \ker u}^\circ$
From Rank Plu... | Let $G$ and $H$ be [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Vector Space|vector spaces]] over a [[Definition:Field (Abstract Algebra)|field]].
Let $\map \LL {G, H}$ be [[Definition:Set of All Linear Transformations|the set of all linear transformations]] from $G$ to $H$.
Let $u \in \map \... | Let $x \in \map \ker u$.
Let $H^*$ be the [[Definition:Algebraic Dual|algebraic dual]] of $H$.
Let $\innerprod x t$ be the [[Definition:Evaluation Linear Transformation/Module Theory|evaluation linear transformation]].
Then:
:$\forall y \in H^*: \innerprod x {\map {u^\intercal} y} = \innerprod {\map u x} y = \inner... | Image of Transpose of Linear Transformation is Annihilator of Kernel | https://proofwiki.org/wiki/Image_of_Transpose_of_Linear_Transformation_is_Annihilator_of_Kernel | https://proofwiki.org/wiki/Image_of_Transpose_of_Linear_Transformation_is_Annihilator_of_Kernel | [
"Linear Algebra"
] | [
"Definition:Dimension of Vector Space",
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Set of All Linear Transformations",
"Definition:Transpose of Linear Transformation",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Annihilator on Algebraic Dual",
"Defin... | [
"Definition:Algebraic Dual",
"Definition:Evaluation Linear Transformation/Module Theory",
"Rank Plus Nullity Theorem",
"Results Concerning Annihilator of Vector Subspace"
] |
proofwiki-18680 | Null Sequence Test for Convergence | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a convergent sequence in $\struct {R, \norm {\,\cdot\,} }$ with limit $l$.
Let $\sequence {y_n}$ be a sequence.
Then:
:$\sequence {y_n}$ converges to the limit $l$ {{iff}} the sequence $\sequence {y_n - x_n}$ is a null sequence | === Necessary Condition ===
Let $\sequence {y_n}$ converge to the limit $l$.
From Difference Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} y_n - x_n = l - l = 0$
Hence $\sequence {y_n - x_n}$ is a null sequence by definition.
{{qed|lemma}} | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence]] in $\struct {R, \norm {\,\cdot\,} }$ with [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l$.
Le... | === Necessary Condition ===
Let $\sequence {y_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converge]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l$.
From [[Difference Rule for Sequences in Normed Division Ring]]:
:$\ds \lim_{n \mathop \to \infty} y_n - x_n = l - l = 0$
Henc... | Null Sequence Test for Convergence | https://proofwiki.org/wiki/Null_Sequence_Test_for_Convergence | https://proofwiki.org/wiki/Null_Sequence_Test_for_Convergence | [
"Convergence",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Sequence",
"Defi... | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Null Sequence/... |
proofwiki-18681 | Ring of Linear Operators is Ring | Let $R$ be a ring.
Let $\struct {G, +}$ be an abelian group..
Let $\struct {G, +, \circ}_R$ be an $R$-module.
Let $\struct {\map {\LL_R} G, +, \circ}$ be the ring of linear operators on $G$, where:
:$+$ denotes pointwise addition
:$\circ$ denotes composition of linear operators.
Then $\struct {\map {\LL_R} G, +, \circ}... | Let $\phi$ and $\psi$ be elements of $\map {\LL_R} G$.
From Composite of R-Algebraic Structure Homomorphisms is Homomorphism, $\phi \circ \psi$ is also an element of $\map {\LL_R} G$.
That is, $\struct {\map {\LL_R} G, \circ}$ is closed.
From Set of Linear Transformations under Pointwise Addition forms Abelian Group, $... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +}$ be an [[Definition:Abelian Group|abelian group]]..
Let $\struct {G, +, \circ}_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $\struct {\map {\LL_R} G, +, \circ}$ be the [[Definition:Ring of Linear Operators|ring of linear opera... | Let $\phi$ and $\psi$ be [[Definition:Element|elements]] of $\map {\LL_R} G$.
From [[Composite of Homomorphisms is Homomorphism/R-Algebraic Structure|Composite of R-Algebraic Structure Homomorphisms is Homomorphism]], $\phi \circ \psi$ is also an [[Definition:Element|element]] of $\map {\LL_R} G$.
That is, $\struct {... | Ring of Linear Operators is Ring | https://proofwiki.org/wiki/Ring_of_Linear_Operators_is_Ring | https://proofwiki.org/wiki/Ring_of_Linear_Operators_is_Ring | [
"Ring of Linear Operators"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Module over Ring",
"Definition:Ring of Linear Operators",
"Definition:Pointwise Addition of Linear Transformations",
"Definition:Composition of Mappings",
"Definition:Linear Operator",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Element",
"Composite of Homomorphisms is Homomorphism/R-Algebraic Structure",
"Definition:Element",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Set of Linear Transformations under Pointwise Addition forms Abelian Group",
"Definition:Abelian Group",
"One-Step Subgroup Test",... |
proofwiki-18682 | Distributional Derivative on Distributions is Linear Operator | The Distributional derivative on Schwartz distributions is a linear operator. | Let $\phi, \psi \in \map \DD \R$ be test functions.
Let $\alpha \in \C$ be a complex number.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Then:
{{begin-eqn}}
{{eqn | l = \map {T'} {\phi + \psi}
| r = - \map T {\paren {\phi + \psi}'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = - \map T... | The [[Definition:Distributional Derivative|Distributional derivative]] on [[Definition:Schwartz Distribution|Schwartz distributions]] is a [[Definition:Linear Operator|linear operator]]. | Let $\phi, \psi \in \map \DD \R$ be [[Definition:Test Function|test functions]].
Let $\alpha \in \C$ be a [[Definition:Complex Number|complex number]].
Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Then:
{{begin-eqn}}
{{eqn | l = \map {T'} {\phi + \psi}
| r = - \m... | Distributional Derivative on Distributions is Linear Operator | https://proofwiki.org/wiki/Distributional_Derivative_on_Distributions_is_Linear_Operator | https://proofwiki.org/wiki/Distributional_Derivative_on_Distributions_is_Linear_Operator | [
"Distributional Derivatives",
"Linear Operators"
] | [
"Definition:Distributional Derivative",
"Definition:Schwartz Distribution",
"Definition:Linear Operator"
] | [
"Definition:Test Function",
"Definition:Complex Number",
"Definition:Schwartz Distribution",
"Sum Rule for Derivatives",
"Definition:Distributional Derivative",
"Definition:Linear Operator"
] |
proofwiki-18683 | Distributional Derivative on Distributions is Continuous Operator | The distributional derivative on Schwartz distributions is a continuous operator. | Let $\mathbf 0$ be the zero mapping.
Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \DD \R$ be a sequence of test functions.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ converge to $\mathbf 0$ in the test function space:
:$\phi_n \stackrel \DD {\longrightarrow} \mathbf 0$
By definition, a test function is a smooth... | The [[Definition:Distributional Derivative|distributional derivative]] on [[Definition:Schwartz Distribution|Schwartz distributions]] is a [[Definition:Continuous Operator|continuous operator]]. | Let $\mathbf 0$ be the [[Definition:Zero Mapping|zero mapping]].
Let $\sequence {\phi_n}_{n \mathop \in \N} \in \map \DD \R$ be a [[Definition:Sequence|sequence]] of [[Definition:Test Function|test functions]].
Let $\sequence {\phi_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Test Function Space|converg... | Distributional Derivative on Distributions is Continuous Operator | https://proofwiki.org/wiki/Distributional_Derivative_on_Distributions_is_Continuous_Operator | https://proofwiki.org/wiki/Distributional_Derivative_on_Distributions_is_Continuous_Operator | [
"Distributional Derivatives",
"Continuous Operators"
] | [
"Definition:Distributional Derivative",
"Definition:Schwartz Distribution",
"Definition:Continuous Operator"
] | [
"Definition:Zero Mapping",
"Definition:Sequence",
"Definition:Test Function",
"Definition:Convergent Sequence/Test Function Space",
"Definition:Test Function Space",
"Definition:Test Function",
"Definition:Smooth Real Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Cont... |
proofwiki-18684 | Composition of Linear Transformations is Isomorphic to Matrix Product | Let $R$ be a ring with unity.
Let $F$, $G$ and $H$ be free $R$-modules of finite dimension $p, n, m > 0$ respectively.
Let $\sequence {a_p}$, $\sequence {b_n}$ and $\sequence {c_m}$ be ordered bases
Let $\map {\LL_R} {G, H}$ denote the set of all linear transformations from $G$ to $H$.
Let $\map {\MM_R} {m, n}$ be the... | Follows directly from Relative Matrix of Composition of Linear Transformations.
{{qed}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $F$, $G$ and $H$ be [[Definition:Free Module over Ring|free $R$-modules]] of [[Definition:Finite Dimensional Free Module|finite dimension]] $p, n, m > 0$ respectively.
Let $\sequence {a_p}$, $\sequence {b_n}$ and $\sequence {c_m}$ be [[Definition:Ordere... | Follows directly from [[Relative Matrix of Composition of Linear Transformations]].
{{qed}} | Composition of Linear Transformations is Isomorphic to Matrix Product | https://proofwiki.org/wiki/Composition_of_Linear_Transformations_is_Isomorphic_to_Matrix_Product | https://proofwiki.org/wiki/Composition_of_Linear_Transformations_is_Isomorphic_to_Matrix_Product | [
"Linear Transformations",
"Matrix Algebra"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension of Module/Finite",
"Definition:Ordered Basis",
"Definition:Set of All Linear Transformations",
"Definition:Matrix Space",
"Definition:Relative Matrix of Linear Transformation"
] | [
"Relative Matrix of Composition of Linear Transformations"
] |
proofwiki-18685 | Slope of Normal is Minus Reciprocal of Tangent | Let $C$ be a curve defined by a real function which is differentiable.
Let $P$ be a point on $C$.
Let the curvature of $C$ at $P$ be non-zero.
Let $r$ be the slope of the tangent to $C$ at $P$.
Let $s$ be the slope of the normal to $C$ at $P$.
Then:
:$r = -\dfrac 1 s$ | By definition, the normal to $C$ at $P$ is defined as being perpendicular to the tangent at $P$ and in the same plane as $P$.
The result follows from Condition for Straight Lines in Plane to be Perpendicular.
{{qed}}
Category:Normals to Curves
Category:Tangents
Category:Analytic Geometry
4ptbzdnli3ckmuyjv65jwd9dbvq6uvn | Let $C$ be a [[Definition:Curve|curve]] defined by a [[Definition:Real Function|real function]] which is [[Definition:Differentiable Real Function|differentiable]].
Let $P$ be a [[Definition:Point|point]] on $C$.
Let the [[Definition:Curvature|curvature]] of $C$ at $P$ be non-[[Definition:Zero (Number)|zero]].
Let $... | By definition, the [[Definition:Normal to Curve|normal]] to $C$ at $P$ is defined as being [[Definition:Perpendicular|perpendicular]] to the [[Definition:Tangent Line|tangent]] at $P$ and in the same [[Definition:The Plane|plane]] as $P$.
The result follows from [[Condition for Straight Lines in Plane to be Perpendicu... | Slope of Normal is Minus Reciprocal of Tangent | https://proofwiki.org/wiki/Slope_of_Normal_is_Minus_Reciprocal_of_Tangent | https://proofwiki.org/wiki/Slope_of_Normal_is_Minus_Reciprocal_of_Tangent | [
"Normals to Curves",
"Tangents",
"Analytic Geometry"
] | [
"Definition:Line/Curve",
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Point",
"Definition:Curvature",
"Definition:Zero (Number)",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Slope/Straight Line",
"Definition:Normal to Curve... | [
"Definition:Normal to Curve",
"Definition:Right Angle/Perpendicular",
"Definition:Tangent Line",
"Definition:Plane Surface/The Plane",
"Condition for Straight Lines in Plane to be Perpendicular",
"Category:Normals to Curves",
"Category:Tangents",
"Category:Analytic Geometry"
] |
proofwiki-18686 | Change of Basis Matrix under Linear Transformation | Let $G$ and $H$ be free unitary $R$-modules of finite dimensions $n, m > 0$ respectively.
Let $\sequence {a_n}$ and $\sequence { {a_n}'}$ be ordered bases of $G$.
Let $\sequence {b_m}$ and $\sequence { {b_m}'}$ be ordered bases of $H$.
Let $u: G \to H$ be a linear transformation.
Let $\sqbrk {u; \sequence {b_m}, \seque... | We have $u = I_H \circ u \circ I_G$
and $\mathbf Q^{-1} = \sqbrk {I_H; \sequence { {b_m}'}, \sequence {b_m} }$.
Thus by Set of Linear Transformations is Isomorphic to Matrix Space:
{{begin-eqn}}
{{eqn | l = \mathbf Q^{-1} \mathbf A \mathbf P
| r = \sqbrk {I_H; \sequence { {b_m}'}, \sequence {b_m} } \sqbrk {u; \se... | Let $G$ and $H$ be [[Definition:Free Module over Ring|free]] [[Definition:Unitary Module over Ring|unitary $R$-modules]] of [[Definition:Dimension of Module|finite dimensions]] $n, m > 0$ respectively.
Let $\sequence {a_n}$ and $\sequence { {a_n}'}$ be [[Definition:Ordered Basis|ordered bases]] of $G$.
Let $\sequence... | We have $u = I_H \circ u \circ I_G$
and $\mathbf Q^{-1} = \sqbrk {I_H; \sequence { {b_m}'}, \sequence {b_m} }$.
Thus by [[Set of Linear Transformations is Isomorphic to Matrix Space]]:
{{begin-eqn}}
{{eqn | l = \mathbf Q^{-1} \mathbf A \mathbf P
| r = \sqbrk {I_H; \sequence { {b_m}'}, \sequence {b_m} } \sqbrk ... | Change of Basis Matrix under Linear Transformation | https://proofwiki.org/wiki/Change_of_Basis_Matrix_under_Linear_Transformation | https://proofwiki.org/wiki/Change_of_Basis_Matrix_under_Linear_Transformation | [
"Change of Basis Matrix under Linear Transformation",
"Linear Algebra",
"Change of Basis"
] | [
"Definition:Free Module over Ring",
"Definition:Unitary Module over Ring",
"Definition:Dimension of Module",
"Definition:Ordered Basis",
"Definition:Ordered Basis",
"Definition:Linear Transformation",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Change of Basis Matrix",
"Defini... | [
"Set of Linear Transformations is Isomorphic to Matrix Space"
] |
proofwiki-18687 | Change of Basis Matrix under Linear Transformation/Converse | Let $G$ and $H$ be free unitary $R$-modules of finite dimensions $n, m > 0$ respectively.
Let $\sequence {a_n}$ be an ordered basis of $G$.
Let $\sequence {b_m}$ be an ordered basis of $H$.
Let $\mathbf A$ and $\mathbf B$ be $m \times n$ matrices over $R$.
Let there exist:
:a nonsingular matrix $\mathbf P$ of order $n$... | Let:
:$\mathbf P = \sqbrk \alpha_n$
:$\mathbf Q = \sqbrk \beta_m$
Let:
:$\forall j \in \closedint 1 n: {a_j}' = \ds \sum_{i \mathop = 1}^n \alpha_{i j} a_i$
:$\forall j \in \closedint 1 m: {b_j}' = \ds \sum_{i \mathop = 1}^m \beta_{i j} b_i$
Then by Invertible Matrix Corresponds with Change of Basis:
:$\sequence { {a_n... | Let $G$ and $H$ be [[Definition:Free Module over Ring|free]] [[Definition:Unitary Module over Ring|unitary $R$-modules]] of [[Definition:Dimension of Module|finite dimensions]] $n, m > 0$ respectively.
Let $\sequence {a_n}$ be an [[Definition:Ordered Basis|ordered basis]] of $G$.
Let $\sequence {b_m}$ be an [[Definit... | Let:
:$\mathbf P = \sqbrk \alpha_n$
:$\mathbf Q = \sqbrk \beta_m$
Let:
:$\forall j \in \closedint 1 n: {a_j}' = \ds \sum_{i \mathop = 1}^n \alpha_{i j} a_i$
:$\forall j \in \closedint 1 m: {b_j}' = \ds \sum_{i \mathop = 1}^m \beta_{i j} b_i$
Then by [[Invertible Matrix Corresponds with Change of Basis]]:
:$\sequence... | Change of Basis Matrix under Linear Transformation/Converse | https://proofwiki.org/wiki/Change_of_Basis_Matrix_under_Linear_Transformation/Converse | https://proofwiki.org/wiki/Change_of_Basis_Matrix_under_Linear_Transformation/Converse | [
"Change of Basis Matrix under Linear Transformation"
] | [
"Definition:Free Module over Ring",
"Definition:Unitary Module over Ring",
"Definition:Dimension of Module",
"Definition:Ordered Basis",
"Definition:Ordered Basis",
"Definition:Matrix",
"Definition:Nonsingular Matrix",
"Definition:Nonsingular Matrix",
"Definition:Linear Transformation",
"Definitio... | [
"Invertible Matrix Corresponds with Change of Basis",
"Definition:Ordered Basis",
"Definition:Change of Basis Matrix",
"Definition:Change of Basis Matrix",
"Definition:Change of Basis Matrix",
"Definition:Set of All Linear Transformations",
"Set of Linear Transformations is Isomorphic to Matrix Space",
... |
proofwiki-18688 | P-adic Expansion of P-adic Unit | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\Z_p$ be the $p$-adic integers.
Let $a \in \Z_p$.
Let $\ldots a_n \ldots a_3 a_2 a_1 a_0$ be the canonical expansion of $a$.
Then:
:$a$ is a $p$-adic unit {{iff}} $a_0 \ne 0$ | From P-adic Unit has Norm Equal to One:
:a is a $p$-adic unit {{iff}} $\norm a_p = 1 = p^0$
By definition of the canonical expansion:
:$a$ is the limit of the $p$-adic expansion $\ds \sum_{n \mathop = 0}^\infty a_n p^n$
From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
:$\norm a_p = p^0$ ... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $a \in \Z_p$.
Let $\ldots a_n \ldots a_3 a_2 a_1 a_0$ be the [[Definition:Canonical P-... | From [[P-adic Unit has Norm Equal to One]]:
:a is a [[Definition:P-adic Unit|$p$-adic unit]] {{iff}} $\norm a_p = 1 = p^0$
By definition of the [[Definition:Canonical P-adic Expansion|canonical expansion]]:
:$a$ is the [[Definition:Limit of P-adic Sequence|limit]] of the [[Definition:P-adic Expansion|$p$-adic expansi... | P-adic Expansion of P-adic Unit | https://proofwiki.org/wiki/P-adic_Expansion_of_P-adic_Unit | https://proofwiki.org/wiki/P-adic_Expansion_of_P-adic_Unit | [
"P-adic Units"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer",
"Definition:Canonical P-adic Expansion",
"Definition:P-adic Unit"
] | [
"P-adic Unit has Norm Equal to One",
"Definition:P-adic Unit",
"Definition:Canonical P-adic Expansion",
"Definition:Limit of Sequence/P-adic Numbers",
"Definition:P-adic Expansion",
"P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient"
] |
proofwiki-18689 | Alternating Group is Simple except on 4 Letters/Lemma 3 | Let $\rho \in S_n$ be an arbitrary $3$-cycle.
Let $\N_n$ denote the initial segment of the natural numbers $\set {0, 1, \ldots, n - 1}$.
Let $i, j, k \in \N_n$ be such that $\rho = \tuple {i, j, k}$.
Then there exists an even permutation $\sigma \in A_n$ such that $\map \sigma 1 = i$, $\map \sigma 2 = j$ and $\map \sig... | We will proceed by cases.
We have that $\card {\set {1, 2, 3} \cap \set {i, j, k} }$ is either $0$, $1$, $2$ or $3$.
;Case $1$: $\card {\set {1, 2, 3} \cap \set {i, j, k} } = 0$ (this case is only possible when $n \ge 6$).
The permutation $\sigma = \tuple {1, i, 2, j} \tuple {3, k}$ is even (by Parity of K-Cycle and Si... | Let $\rho \in S_n$ be an arbitrary [[Definition:Cyclic Permutation|$3$-cycle]].
Let $\N_n$ denote the [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]] $\set {0, 1, \ldots, n - 1}$.
Let $i, j, k \in \N_n$ be such that $\rho = \tuple {i, j, k}$.
Then there exists an [[Definition... | We will proceed [[Proof by Cases|by cases]].
We have that $\card {\set {1, 2, 3} \cap \set {i, j, k} }$ is either $0$, $1$, $2$ or $3$.
;Case $1$: $\card {\set {1, 2, 3} \cap \set {i, j, k} } = 0$ (this case is only possible when $n \ge 6$).
The [[Definition:Permutation on n Letters|permutation]] $\sigma = \tuple {... | Alternating Group is Simple except on 4 Letters/Lemma 3 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_3 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_3 | [
"Alternating Group is Simple except on 4 Letters"
] | [
"Definition:Cyclic Permutation",
"Definition:Initial Segment of Natural Numbers",
"Definition:Even Permutation"
] | [
"Proof by Cases",
"Definition:Permutation on n Letters",
"Definition:Even Permutation",
"Parity of K-Cycle",
"Sign of Composition of Permutations",
"Definition:Permutation on n Letters",
"Definition:Even Permutation",
"Parity of K-Cycle",
"Sign of Composition of Permutations",
"Definition:Permutat... |
proofwiki-18690 | Characterization of Rational P-adic Integer | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\Q$ be the rational numbers.
Then:
:$\Z_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid b}$ | Let $\norm{\,\cdot\,}^\Q _p$ denote the $p$-adic norm on the rational numbers.
We have:
{{begin-eqn}}
{{eqn | l = \Z_p \cap \Q
| r = \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}
| c = {{Defof|P-adic Integer|$p$-adic integers}}
}}
{{eqn | r = \set{\dfrac a b \in \Q : \norm{\dfrac a b}^\Q_p \le 1}
... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Q$ be the [[Definition:Rational Number... | Let $\norm{\,\cdot\,}^\Q _p$ denote the [[Definition:P-adic Norm on Rational Numbers|$p$-adic norm]] on the [[Definition:Rational Number|rational numbers]].
We have:
{{begin-eqn}}
{{eqn | l = \Z_p \cap \Q
| r = \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}
| c = {{Defof|P-adic Integer|$p$-adic integ... | Characterization of Rational P-adic Integer | https://proofwiki.org/wiki/Characterization_of_Rational_P-adic_Integer | https://proofwiki.org/wiki/Characterization_of_Rational_P-adic_Integer | [
"P-adic Integers"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Rational Number"
] | [
"Definition:P-adic Norm/Rational Numbers",
"Definition:Rational Number",
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Valuation Ring of P-adic Norm on Rationals"
] |
proofwiki-18691 | Characterization of Rational P-adic Unit | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $\Z^\times_p$ be the $p$-adic units.
Let $\Q$ be the rational numbers.
Then:
:$\Z^\times_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid ab}$ | Let $\norm{\,\cdot\,}^\Q _p$ denote the $p$-adic norm on the rational numbers.
We have:
{{begin-eqn}}
{{eqn | l = \Z^\times_p \cap \Q
| r = \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p = 1}
| c = P-adic Unit has Norm Equal to One
}}
{{eqn | r = \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1} \setminus... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Z^\times_p$ be the [[Definition:P-adic Unit|$p$-adic units]].
Let $\Q$ be the [[Definition:Rational Number|rational numbers]].
Then:
:$\Z^\times_p \ca... | Let $\norm{\,\cdot\,}^\Q _p$ denote the [[Definition:P-adic Norm on Rational Numbers|$p$-adic norm]] on the [[Definition:Rational Number|rational numbers]].
We have:
{{begin-eqn}}
{{eqn | l = \Z^\times_p \cap \Q
| r = \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p = 1}
| c = [[P-adic Unit has Norm Equal to... | Characterization of Rational P-adic Unit | https://proofwiki.org/wiki/Characterization_of_Rational_P-adic_Unit | https://proofwiki.org/wiki/Characterization_of_Rational_P-adic_Unit | [
"P-adic Integers"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Unit",
"Definition:Rational Number"
] | [
"Definition:P-adic Norm/Rational Numbers",
"Definition:Rational Number",
"P-adic Unit has Norm Equal to One",
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Valuation Ideal of P-adic Norm on Rationals",
"Valuation Ring of P-adic Norm on Rationals",
"Divisors of Product of Coprime Integers/Cor... |
proofwiki-18692 | Equivalence of Definitions of Matrix Equivalence | Let $R$ be a ring with unity.
Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$.
{{TFAE|def = Matrix Equivalence}} | This is specifically demonstrated in Change of Basis Matrix under Linear Transformation.
{{qed}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $\mathbf A, \mathbf B$ be [[Definition:Matrix|$m \times n$ matrices]] over $R$.
{{TFAE|def = Matrix Equivalence}} | This is specifically demonstrated in [[Change of Basis Matrix under Linear Transformation]].
{{qed}} | Equivalence of Definitions of Matrix Equivalence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matrix_Equivalence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matrix_Equivalence | [
"Matrix Equivalence"
] | [
"Definition:Ring with Unity",
"Definition:Matrix"
] | [
"Change of Basis Matrix under Linear Transformation"
] |
proofwiki-18693 | Equivalence of Definitions of Matrix Similarity | Let $R$ be a ring with unity.
Let $n \in \N_{>0}$ be a natural number.
Let $\mathbf A, \mathbf B$ be square matrices of order $n$ over $R$.
{{TFAE|def = Matrix Similarity}} | This is specifically demonstrated in {{Corollary|Change of Basis Matrix under Linear Transformation}}.
{{qed}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $n \in \N_{>0}$ be a [[Definition:Natural Number|natural number]].
Let $\mathbf A, \mathbf B$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Matrix|order]] $n$ over $R$.
{{TFAE|def = Matrix Similarity}} | This is specifically demonstrated in {{Corollary|Change of Basis Matrix under Linear Transformation}}.
{{qed}} | Equivalence of Definitions of Matrix Similarity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matrix_Similarity | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matrix_Similarity | [
"Matrix Similarity"
] | [
"Definition:Ring with Unity",
"Definition:Natural Numbers",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Order"
] | [] |
proofwiki-18694 | Multiple Function on Ring is Homomorphism | :$g_a$ is a group homomorphism from $\struct {\Z, +}$ to $\struct {R, +}$. | {{begin-eqn}}
{{eqn | l = \map {g_a} m + \map {g_a} n
| r = m \cdot a + n \cdot a
| c = {{Defof|Integral Multiple|subdef = Rings and Fields}}
}}
{{eqn | r = \paren {m + n} \cdot a
| c = Integral Multiple Distributes over Ring Addition
}}
{{eqn | r = \map {g_a} {m + n}
| c = {{Defof|Integral Mult... | :$g_a$ is a [[Definition:Group Homomorphism|group homomorphism]] from $\struct {\Z, +}$ to $\struct {R, +}$. | {{begin-eqn}}
{{eqn | l = \map {g_a} m + \map {g_a} n
| r = m \cdot a + n \cdot a
| c = {{Defof|Integral Multiple|subdef = Rings and Fields}}
}}
{{eqn | r = \paren {m + n} \cdot a
| c = [[Integral Multiple Distributes over Ring Addition]]
}}
{{eqn | r = \map {g_a} {m + n}
| c = {{Defof|Integral ... | Multiple Function on Ring is Homomorphism | https://proofwiki.org/wiki/Multiple_Function_on_Ring_is_Homomorphism | https://proofwiki.org/wiki/Multiple_Function_on_Ring_is_Homomorphism | [
"Group Homomorphisms",
"Integers",
"Ring Theory"
] | [
"Definition:Group Homomorphism"
] | [
"Integral Multiple Distributes over Ring Addition"
] |
proofwiki-18695 | Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function | :$\ideal p \subseteq \map \ker {g_a}$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is the principal ideal of $\Z$ generated by $p$. | We have from Multiple Function on Ring is Homomorphism that $g_a$ is a group homomorphism.
By definition of kernel:
:$x \in \map \ker {g_a} \iff \map {g_a} x = 0_R$
Hence to show that $\ideal p \subseteq \map \ker {g_a}$, we need to show that:
:$\forall x \in \ideal p: \map {g_a} x = 0_R$
By definition of '''characteri... | :$\ideal p \subseteq \map \ker {g_a}$
where:
:$\map \ker {g_a}$ is the [[Definition:Kernel of Group Homomorphism|kernel]] of $g_a$
:$\ideal p$ is the [[Definition:Principal Ideal of Ring|principal ideal]] of $\Z$ generated by $p$. | We have from [[Multiple Function on Ring is Homomorphism]] that $g_a$ is a [[Definition:Group Homomorphism|group homomorphism]].
By definition of [[Definition:Kernel of Group Homomorphism|kernel]]:
:$x \in \map \ker {g_a} \iff \map {g_a} x = 0_R$
Hence to show that $\ideal p \subseteq \map \ker {g_a}$, we need to sho... | Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function | https://proofwiki.org/wiki/Principal_Ideal_of_Characteristic_of_Ring_is_Subset_of_Kernel_of_Multiple_Function | https://proofwiki.org/wiki/Principal_Ideal_of_Characteristic_of_Ring_is_Subset_of_Kernel_of_Multiple_Function | [
"Homomorphism from Integers into Ring with Unity"
] | [
"Definition:Kernel of Group Homomorphism",
"Definition:Principal Ideal of Ring"
] | [
"Multiple Function on Ring is Homomorphism",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Characteristic of Ring",
"Definition:Kernel of Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Multiple of Ring Product"
] |
proofwiki-18696 | Multiplication Function on Ring with Unity is Zero if Characteristic is Divisor | :$p \divides n \implies n \cdot a = 0_R$
where $p \divides n$ denotes that $p$ is a divisor of $n$. | Let $p > 0$.
From Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function:
:$\ideal p \subseteq \map \ker {g_a}$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is the principal ideal of $\Z$ generated by $p$.
We have:
{{begin-eqn}}
{{eqn | q =
| l = p
| o = \divides... | :$p \divides n \implies n \cdot a = 0_R$
where $p \divides n$ denotes that $p$ is a [[Definition:Divisor of Integer|divisor]] of $n$. | Let $p > 0$.
From [[Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function]]:
:$\ideal p \subseteq \map \ker {g_a}$
where:
:$\map \ker {g_a}$ is the [[Definition:Kernel of Group Homomorphism|kernel]] of $g_a$
:$\ideal p$ is the [[Definition:Principal Ideal of Ring|principal ideal]] of $\Z$... | Multiplication Function on Ring with Unity is Zero if Characteristic is Divisor | https://proofwiki.org/wiki/Multiplication_Function_on_Ring_with_Unity_is_Zero_if_Characteristic_is_Divisor | https://proofwiki.org/wiki/Multiplication_Function_on_Ring_with_Unity_is_Zero_if_Characteristic_is_Divisor | [
"Homomorphism from Integers into Ring with Unity"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function",
"Definition:Kernel of Group Homomorphism",
"Definition:Principal Ideal of Ring",
"Integral Ideal iff Set of Integer Multiples",
"Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function"
] |
proofwiki-18697 | Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic | Let $a \in R$ such that $a$ is not a zero divisor of $R$.
Then:
:$\map \ker {g_a} = \ideal p$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is the principal ideal of $\Z$ generated by $p$. | From Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function we have:
:$\ideal p \subseteq \map \ker {g_a}$
for all $a \in R$.
It remains to be shown that for all $a \in R$ such that $a$ is not a zero divisor of $R$:
:$\map \ker {g_a} \subseteq \ideal p$
So:
{{begin-eqn}}
{{eqn | l = n
... | Let $a \in R$ such that $a$ is not a [[Definition:Zero Divisor of Ring|zero divisor]] of $R$.
Then:
:$\map \ker {g_a} = \ideal p$
where:
:$\map \ker {g_a}$ is the [[Definition:Kernel of Group Homomorphism|kernel]] of $g_a$
:$\ideal p$ is the [[Definition:Principal Ideal of Ring|principal ideal]] of $\Z$ generated by $... | From [[Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function]] we have:
:$\ideal p \subseteq \map \ker {g_a}$
for all $a \in R$.
It remains to be shown that for all $a \in R$ such that $a$ is not a [[Definition:Zero Divisor of Ring|zero divisor]] of $R$:
:$\map \ker {g_a} \subseteq \ideal... | Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic | https://proofwiki.org/wiki/Kernel_of_Non-Zero_Divisor_Multiple_Function_is_Primary_Ideal_of_Characteristic | https://proofwiki.org/wiki/Kernel_of_Non-Zero_Divisor_Multiple_Function_is_Primary_Ideal_of_Characteristic | [
"Homomorphism from Integers into Ring with Unity"
] | [
"Definition:Zero Divisor/Ring",
"Definition:Kernel of Group Homomorphism",
"Definition:Principal Ideal of Ring"
] | [
"Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function",
"Definition:Zero Divisor/Ring",
"Multiple of Ring Product",
"Multiple of Ring Product",
"Definition:Zero Divisor/Ring",
"Definition:Subset"
] |
proofwiki-18698 | Kernel of Multiple Function on Ring with Characteristic Zero is Trivial | Let $a \in R$ such that $a$ is not a zero divisor of $R$.
Let the characteristic of $R$ be $0$.
Then:
:$\map \ker {g_a} = \set {0_R}$
where $\ker$ denotes the kernel of $g_a$. | For $n = 0$, we trivially have $n \cdot a = 0_R$.
{{AimForCont}} $\exists n \ne 0: n \cdot a = 0_R$.
Then:
{{begin-eqn}}
{{eqn | l = n
| o = \in
| r = \map \ker {g_a}
| c = {{Defof|Kernel of Group Homomorphism}}
}}
{{eqn | ll= \leadsto
| l = n
| o = \in
| r = \ideal p
| c = Ker... | Let $a \in R$ such that $a$ is not a [[Definition:Zero Divisor of Ring|zero divisor]] of $R$.
Let the [[Definition:Characteristic of Ring|characteristic]] of $R$ be $0$.
Then:
:$\map \ker {g_a} = \set {0_R}$
where $\ker$ denotes the [[Definition:Kernel of Group Homomorphism|kernel]] of $g_a$. | For $n = 0$, we trivially have $n \cdot a = 0_R$.
{{AimForCont}} $\exists n \ne 0: n \cdot a = 0_R$.
Then:
{{begin-eqn}}
{{eqn | l = n
| o = \in
| r = \map \ker {g_a}
| c = {{Defof|Kernel of Group Homomorphism}}
}}
{{eqn | ll= \leadsto
| l = n
| o = \in
| r = \ideal p
| c =... | Kernel of Multiple Function on Ring with Characteristic Zero is Trivial | https://proofwiki.org/wiki/Kernel_of_Multiple_Function_on_Ring_with_Characteristic_Zero_is_Trivial | https://proofwiki.org/wiki/Kernel_of_Multiple_Function_on_Ring_with_Characteristic_Zero_is_Trivial | [
"Homomorphism from Integers into Ring with Unity"
] | [
"Definition:Zero Divisor/Ring",
"Definition:Characteristic of Ring",
"Definition:Kernel of Group Homomorphism"
] | [
"Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic",
"Definition:Characteristic of Ring",
"Definition:Contradiction",
"Definition:Characteristic of Ring",
"Proof by Contradiction"
] |
proofwiki-18699 | Multiple Function on Ring is Zero iff Characteristic is Divisor | Let $a \in R$ such that $a$ is not a zero divisor of $R$.
Then:
:$n \cdot a = 0_R$
{{iff}}:
:$p \divides n$ | Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:
:$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$
Then from Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic:
:$\map \ker {g_a} = \ideal p$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is... | Let $a \in R$ such that $a$ is not a [[Definition:Zero Divisor of Ring|zero divisor]] of $R$.
Then:
:$n \cdot a = 0_R$
{{iff}}:
:$p \divides n$ | Let $g_a: \Z \to R$ be the [[Definition:Mapping|mapping]] from the [[Definition:Integer|integers]] into $R$ defined as:
:$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$
Then from [[Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic]]:
:$\map \ker {g_a} = \ideal p$
where:
:$\ma... | Multiple Function on Ring is Zero iff Characteristic is Divisor | https://proofwiki.org/wiki/Multiple_Function_on_Ring_is_Zero_iff_Characteristic_is_Divisor | https://proofwiki.org/wiki/Multiple_Function_on_Ring_is_Zero_iff_Characteristic_is_Divisor | [
"Homomorphism from Integers into Ring with Unity"
] | [
"Definition:Zero Divisor/Ring"
] | [
"Definition:Mapping",
"Definition:Integer",
"Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic",
"Definition:Kernel of Group Homomorphism",
"Definition:Principal Ideal of Ring",
"Definition:Kernel of Group Homomorphism",
"Definition:Principal Ideal of Ring"
] |
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