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-12300 | Consecutive Integers whose Sums of Squares of Divisors are Equal | The only two consecutive positive integers whose sums of the squares of their divisors are equal are $6$ and $7$. | The divisors of $6$ are
:$1, 2, 3, 6$
and so the sum of the squares of the divisors of $6$ is:
:$1^2 + 2^2 + 3^2 + 6^2 = 1 + 4 + 9 + 36 = 50$
The divisors of $7$ are
:$1, 7$
and so the sum of the squares of the divisors of $7$ is:
:$1^2 + 7^2 = 1 + 49 = 50$
It remains to be shown that there are no more.
Let $n \ge 7$ b... | The only two consecutive [[Definition:Positive Integer|positive integers]] whose sums of the [[Definition:Square (Algebra)|squares]] of their [[Definition:Divisor of Integer|divisors]] are equal are $6$ and $7$. | The [[Definition:Divisor of Integer|divisors]] of $6$ are
:$1, 2, 3, 6$
and so the sum of the [[Definition:Square (Algebra)|squares]] of the [[Definition:Divisor of Integer|divisors]] of $6$ is:
:$1^2 + 2^2 + 3^2 + 6^2 = 1 + 4 + 9 + 36 = 50$
The [[Definition:Divisor of Integer|divisors]] of $7$ are
:$1, 7$
and so the... | Consecutive Integers whose Sums of Squares of Divisors are Equal | https://proofwiki.org/wiki/Consecutive_Integers_whose_Sums_of_Squares_of_Divisors_are_Equal | https://proofwiki.org/wiki/Consecutive_Integers_whose_Sums_of_Squares_of_Divisors_are_Equal | [
"6",
"7"
] | [
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Square/Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Square/Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Square/Function... |
proofwiki-12301 | Digital Root of 3 Consecutive Numbers ending in Multiple of 3 | Let $n$, $n + 1$ and $n + 2$ be positive integers such that $n + 2$ is a multiple of $3$.
Let $m = n + \paren {n + 1} + \paren {n + 2}$.
Then the digital root of $m$ is $6$. | Let $n + 2$ be expressed as $3 r$ for some positive integer $r$.
Then:
{{begin-eqn}}
{{eqn | l = m
| r = \paren {3 r - 2} + \paren {3 r - 1} + 3 r
| c =
}}
{{eqn | r = 9 r - 3
| c =
}}
{{eqn | r = 9 \paren {r - 1} + 6
| c =
}}
{{end-eqn}}
The result follows from Digital Root of Number equals ... | Let $n$, $n + 1$ and $n + 2$ be [[Definition:Positive Integer|positive integers]] such that $n + 2$ is a [[Definition:Multiple of Integer|multiple]] of $3$.
Let $m = n + \paren {n + 1} + \paren {n + 2}$.
Then the [[Definition:Digital Root|digital root]] of $m$ is $6$. | Let $n + 2$ be expressed as $3 r$ for some [[Definition:Positive Integer|positive integer]] $r$.
Then:
{{begin-eqn}}
{{eqn | l = m
| r = \paren {3 r - 2} + \paren {3 r - 1} + 3 r
| c =
}}
{{eqn | r = 9 r - 3
| c =
}}
{{eqn | r = 9 \paren {r - 1} + 6
| c =
}}
{{end-eqn}}
The result follows ... | Digital Root of 3 Consecutive Numbers ending in Multiple of 3 | https://proofwiki.org/wiki/Digital_Root_of_3_Consecutive_Numbers_ending_in_Multiple_of_3 | https://proofwiki.org/wiki/Digital_Root_of_3_Consecutive_Numbers_ending_in_Multiple_of_3 | [
"6",
"Digital Roots"
] | [
"Definition:Positive/Integer",
"Definition:Multiple/Integer",
"Definition:Digital Root"
] | [
"Definition:Positive/Integer",
"Digital Root of Number equals its Excess over Multiple of 9"
] |
proofwiki-12302 | Characteristics of Regular 4-Dimensional Polytopes | The $4$-dimensional regular polytopes have the following characteristics:
{| border="1"
|-
! Name
! No. of cells
! No. of faces
! No. of edges
! No. of vertices
! Dual
|-
| align="right" | Pentatope
| align="right" | $5$
| align="right" | $10$
| align="right" | $10$
| align="right" | $5$
| align="right" | Self-dual
|-
... | {{ProofWanted|Much background work to be covered}} | The [[Definition:Dimension (Geometry)|$4$-dimensional]] [[Definition:Regular Polytope|regular polytopes]] have the following characteristics:
{| border="1"
|-
! Name
! No. of [[Definition:Cell of Polytope|cells]]
! No. of [[Definition:Face of Polytope|faces]]
! No. of [[Definition:Edge of Polytope|edges]]
! No. of [[D... | {{ProofWanted|Much background work to be covered}} | Characteristics of Regular 4-Dimensional Polytopes | https://proofwiki.org/wiki/Characteristics_of_Regular_4-Dimensional_Polytopes | https://proofwiki.org/wiki/Characteristics_of_Regular_4-Dimensional_Polytopes | [
"Polytopes"
] | [
"Definition:Dimension (Geometry)",
"Definition:Regular Polytope",
"Definition:Cell of Polytope",
"Definition:Face of Polytope",
"Definition:Edge of Polytope",
"Definition:Vertex of Polytope",
"Definition:Dual of 4-Polytope",
"Definition:Pentatope",
"Definition:Self-Dual 4-Polytope",
"Definition:Te... | [] |
proofwiki-12303 | Complex Power is of Exponential Order Epsilon | Let:
:$f: \hointr 0 \to \to \C: t \mapsto t^\phi$
be $t$ to the power of $\phi$, for $\phi \in \C$, defined on its principal branch.
Let $\map \Re \phi > -1$.
Then $f$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | {{begin-eqn}}
{{eqn | l = \size {t^\phi}
| r = t^{\map \Re \phi}
| c = Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part
}}
{{end-eqn}}
The result follows from Real Power is of Exponential Order Epsilon.
{{qed}}
Category:Exponential Order
fmibd8v9dkf2rrq46kvp0cbd... | Let:
:$f: \hointr 0 \to \to \C: t \mapsto t^\phi$
be [[Definition:Power to Complex Number|$t$ to the power of $\phi$]], for $\phi \in \C$, defined on its [[Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base|principal branch]].
Let $\map \Re \phi > -1$.
Then $f$ is of [[Definition:Expone... | {{begin-eqn}}
{{eqn | l = \size {t^\phi}
| r = t^{\map \Re \phi}
| c = [[Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part]]
}}
{{end-eqn}}
The result follows from [[Real Power is of Exponential Order Epsilon]].
{{qed}}
[[Category:Exponential Order]]
fmibd8v9dk... | Complex Power is of Exponential Order Epsilon | https://proofwiki.org/wiki/Complex_Power_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Complex_Power_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Power (Algebra)/Complex Number",
"Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base",
"Definition:Exponential Order/Real Index"
] | [
"Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part",
"Real Power is of Exponential Order Epsilon",
"Category:Exponential Order"
] |
proofwiki-12304 | Three Regular Tessellations/Hexagons | Regular hexagons form a regular tessellation:
:400px | Let $s \in \R_{>0}$ be the side length of the regular hexagons.
For all $x, y \in \Z$, let the center $Q_{x,y}$ of each regular hexagon have Cartesian coordinates:
:$Q_{x,y} = s \tuple {x, \sqrt 3 y + \dfrac {\sqrt 3 \paren {3 - m} } 4}$
where:
:$m = \begin{cases} 1 & : \textrm {for $x$ even} \\ -1 & : \textrm{for $x$ ... | [[Definition:Regular Hexagon|Regular hexagons]] form a [[Definition:Regular Tessellation|regular tessellation]]:
:[[File:RegularHexagonTessellation.png|400px]] | Let $s \in \R_{>0}$ be the [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] of the [[Definition:Regular Hexagon|regular hexagons]].
For all $x, y \in \Z$, let the [[Definition:Center of Regular Polygon|center]] $Q_{x,y}$ of each [[Definition:Regular Hexagon|regular hexagon]] have [[Definition:C... | Three Regular Tessellations/Hexagons | https://proofwiki.org/wiki/Three_Regular_Tessellations/Hexagons | https://proofwiki.org/wiki/Three_Regular_Tessellations/Hexagons | [
"Three Regular Tessellations",
"Hexagons"
] | [
"Definition:Hexagon/Regular",
"Definition:Regular Tessellation",
"File:RegularHexagonTessellation.png"
] | [
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Hexagon/Regular",
"Definition:Polygon/Regular/Center",
"Definition:Hexagon/Regular",
"Definition:Cartesian Plane/Ordered Pair",
"Regular Hexagon is composed of Equilateral Triangles",
"Definition:Hexagon",
"Definition:Triangu... |
proofwiki-12305 | Finite Subset Bounds Element of Finite Suprema Set and Lower Closure | Let $L = \struct {S, \vee, \preceq}$ be join semilattice.
Let $I$ be ideal in $L$.
Let $X$ be non empty finite subset of $S$.
Let $x \in S$ such that
:$x \in \paren {\map {\operatorname {finsups} } {F \cup X} }^\preceq$
where
:$\operatorname {finsups}$ denotes the finite suprema set
:$X^\preceq$ denotes the lower closu... | This follows by {{mutatis}} of the proof of Finite Subset Bounds Element of Finite Infima Set and Upper Closure.
{{qed}} | Let $L = \struct {S, \vee, \preceq}$ be [[Definition:Join Semilattice|join semilattice]].
Let $I$ be [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Let $X$ be [[Definition:Non-Empty Set|non empty]] [[Definition:Finite Subset|finite subset]] of $S$.
Let $x \in S$ such that
:$x \in \paren {\map {\operatorname {fins... | This follows by {{mutatis}} of the proof of [[Finite Subset Bounds Element of Finite Infima Set and Upper Closure]].
{{qed}} | Finite Subset Bounds Element of Finite Suprema Set and Lower Closure | https://proofwiki.org/wiki/Finite_Subset_Bounds_Element_of_Finite_Suprema_Set_and_Lower_Closure | https://proofwiki.org/wiki/Finite_Subset_Bounds_Element_of_Finite_Suprema_Set_and_Lower_Closure | [
"Join and Meet Semilattices"
] | [
"Definition:Join Semilattice",
"Definition:Ideal (Order Theory)",
"Definition:Non-Empty Set",
"Definition:Finite Subset",
"Definition:Finite Suprema Set",
"Definition:Lower Closure/Set"
] | [
"Finite Subset Bounds Element of Finite Infima Set and Upper Closure"
] |
proofwiki-12306 | Fermat Quotient of 2 wrt p is Square iff p is 3 or 7 | Let $p$ be a prime number.
The Fermat quotient of $2$ with respect to $p$:
:$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$
is a square {{iff}} $p = 3$ or $p = 7$. | When $p = 3$:
:$\map {q_3} 2 = \dfrac {2^{3 - 1} - 1} 3 = 1$
which is square.
When $p = 7$:
:$\map {q_7} 2 = \dfrac {2^{7 - 1} - 1} 7 = \dfrac {63} 7 = 9$
which is square.
To show that these are the only ones, we observe that since $p$ is an odd prime, write:
:$p = 2 n + 1$ for $n \ge 1$.
Let $\map {q_p} 2$ be a square... | Let $p$ be a [[Definition:Prime Number|prime number]].
The [[Definition:Fermat Quotient|Fermat quotient]] of $2$ with respect to $p$:
:$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$
is a [[Definition:Square Number|square]] {{iff}} $p = 3$ or $p = 7$. | When $p = 3$:
:$\map {q_3} 2 = \dfrac {2^{3 - 1} - 1} 3 = 1$
which is [[Definition:Square Number|square]].
When $p = 7$:
:$\map {q_7} 2 = \dfrac {2^{7 - 1} - 1} 7 = \dfrac {63} 7 = 9$
which is [[Definition:Square Number|square]].
To show that these are the only ones, we observe that since $p$ is an [[Definition:Odd ... | Fermat Quotient of 2 wrt p is Square iff p is 3 or 7 | https://proofwiki.org/wiki/Fermat_Quotient_of_2_wrt_p_is_Square_iff_p_is_3_or_7 | https://proofwiki.org/wiki/Fermat_Quotient_of_2_wrt_p_is_Square_iff_p_is_3_or_7 | [
"Fermat Quotients"
] | [
"Definition:Prime Number",
"Definition:Fermat Quotient",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Square Number",
"Definition:Odd Prime",
"Definition:Square Number",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-12307 | Divisibility of Elements of Pythagorean Triple by 7 | Let $\tuple {a, b, c}$ be a Pythagorean triple such that $a^2 + b^2 = c^2$.
Then at least one of $a$, $b$, $a + b$ or $a - b$ is divisible by $7$. | It is sufficient to consider primitive Pythagorean triples.
From Solutions of Pythagorean Equation, the set of all Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
: $m, n \in \Z_{>0}$ are (strictly) positive integers
: $m \perp n$, that is, $m$ and $n$ are coprime
: $m$ and $n$ are o... | Let $\tuple {a, b, c}$ be a [[Definition:Pythagorean Triple|Pythagorean triple]] such that $a^2 + b^2 = c^2$.
Then at least one of $a$, $b$, $a + b$ or $a - b$ is [[Definition:Divisor of Integer|divisible]] by $7$. | It is sufficient to consider [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triples]].
From [[Solutions of Pythagorean Equation]], the [[Definition:Set|set]] of all [[Definition:Pythagorean Triple|Pythagorean triples]] is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
: $m, n \in \Z_{>0}... | Divisibility of Elements of Pythagorean Triple by 7 | https://proofwiki.org/wiki/Divisibility_of_Elements_of_Pythagorean_Triple_by_7 | https://proofwiki.org/wiki/Divisibility_of_Elements_of_Pythagorean_Triple_by_7 | [
"Pythagorean Triples",
"Divisors",
"7"
] | [
"Definition:Pythagorean Triple",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Pythagorean Triple/Primitive",
"Solutions of Pythagorean Equation",
"Definition:Set",
"Definition:Pythagorean Triple",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Parity of Integer",
"Definition:Congruence (Number Theory)/Integers",
"Modulo Addition... |
proofwiki-12308 | Smaller Elements of Pythagorean Triple not both Odd | Let $\left({x, y, z}\right)$ be a Pythagorean triple, i.e. integers such that $x^2 + y^2 = z^2$.
Then $x$ and $y$ cannot both be odd. | {{AimForCont}} $x$ and $y$ are both odd such that:
:$\exists z \in \Z: x^2 + y^2 = z^2$
Then:
:$x^2 + y^2 \equiv 1 + 1 \equiv 2 \pmod 4$
But from Square Modulo 4:
:$z^2 \equiv 0 \pmod 4$ or $z^2 \equiv 1 \pmod 4$
Thus $x^2 + y^2$ can not be square.
It follows by Proof by Contradiction that $x$ and $y$ cannot both be o... | Let $\left({x, y, z}\right)$ be a [[Definition:Pythagorean Triple|Pythagorean triple]], i.e. [[Definition:Integer|integers]] such that $x^2 + y^2 = z^2$.
Then $x$ and $y$ cannot both be [[Definition:Odd Integer|odd]]. | {{AimForCont}} $x$ and $y$ are both [[Definition:Odd Integer|odd]] such that:
:$\exists z \in \Z: x^2 + y^2 = z^2$
Then:
:$x^2 + y^2 \equiv 1 + 1 \equiv 2 \pmod 4$
But from [[Square Modulo 4]]:
:$z^2 \equiv 0 \pmod 4$ or $z^2 \equiv 1 \pmod 4$
Thus $x^2 + y^2$ can not be [[Definition:Square Number|square]].
It foll... | Smaller Elements of Pythagorean Triple not both Odd | https://proofwiki.org/wiki/Smaller_Elements_of_Pythagorean_Triple_not_both_Odd | https://proofwiki.org/wiki/Smaller_Elements_of_Pythagorean_Triple_not_both_Odd | [
"Pythagorean Triples"
] | [
"Definition:Pythagorean Triple",
"Definition:Integer",
"Definition:Odd Integer"
] | [
"Definition:Odd Integer",
"Square Modulo 4",
"Definition:Square Number",
"Proof by Contradiction",
"Definition:Odd Integer",
"Category:Pythagorean Triples"
] |
proofwiki-12309 | Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed | Let $P = \struct {S, \preceq}$ be an ordered set.
Let $A$ be a set of subsets of $S$.
Let
:$\forall X \in A: X$ is directed.
Let
:$\forall X, Y \in A: \exists Z \in A: X \cup Y \subseteq Z$
Then $\bigcup A$ is directed. | Let $x, y \in \bigcup A$.
By definition of union:
:$\exists X \in A: x \in X$
and
:$\exists Y \in A: y \in Y$
By assumption:
:$\exists Z \in A: X \cup Y \subseteq Z$
By definition of union:
:$x, y \in X \cup Y$
By definition of subset:
:$x, y \in Z$
By assumption:
:$Z$ is directed.
By definition of directed subset:
:$\... | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $A$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$.
Let
:$\forall X \in A: X$ is [[Definition:Directed Subset|directed]].
Let
:$\forall X, Y \in A: \exists Z \in A: X \cup Y \subseteq Z$
Then $\bigcup A$ i... | Let $x, y \in \bigcup A$.
By definition of [[Definition:Union of Set of Sets|union]]:
:$\exists X \in A: x \in X$
and
:$\exists Y \in A: y \in Y$
By assumption:
:$\exists Z \in A: X \cup Y \subseteq Z$
By definition of [[Definition:Set Union|union]]:
:$x, y \in X \cup Y$
By definition of [[Definition:Subset|subset]... | Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed | https://proofwiki.org/wiki/Every_Element_is_Directed_and_Every_Two_Elements_are_Included_in_Third_Element_implies_Union_is_Directed | https://proofwiki.org/wiki/Every_Element_is_Directed_and_Every_Two_Elements_are_Included_in_Third_Element_implies_Union_is_Directed | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Directed Subset"
] | [
"Definition:Set Union/Set of Sets",
"Definition:Set Union",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Set Union/Set of Sets",
"Definition:Directed Subset"
] |
proofwiki-12310 | Every Element is Lower implies Union is Lower | Let $\struct {S, \preceq}$ be an ordered set.
Let $A$ be a set of subsets of $S$.
Let
:$\forall X \in A: X$ is a lower section.
Then $\bigcup A$ is a lower section. | Let $x \in \bigcup A, y \in S$ such that:
:$y \preceq x$
By definition of union:
:$\exists X \in A: x \in X$
By assumption:
:$X$ is a lower section.
By definition of lower section:
:$y \in X$
Thus by definition of union:
:$y \in \bigcup A$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $A$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$.
Let
:$\forall X \in A: X$ is a [[Definition:Lower Section|lower section]].
Then $\bigcup A$ is a [[Definition:Lower Section|lower section]]. | Let $x \in \bigcup A, y \in S$ such that:
:$y \preceq x$
By definition of [[Definition:Union of Set of Sets|union]]:
:$\exists X \in A: x \in X$
By assumption:
:$X$ is a [[Definition:Lower Section|lower section]].
By definition of [[Definition:Lower Section|lower section]]:
:$y \in X$
Thus by definition of [[Defini... | Every Element is Lower implies Union is Lower | https://proofwiki.org/wiki/Every_Element_is_Lower_implies_Union_is_Lower | https://proofwiki.org/wiki/Every_Element_is_Lower_implies_Union_is_Lower | [
"Lower Sections"
] | [
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Lower Section",
"Definition:Lower Section"
] | [
"Definition:Set Union/Set of Sets",
"Definition:Lower Section",
"Definition:Lower Section",
"Definition:Set Union/Set of Sets"
] |
proofwiki-12311 | If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.
Let $I$ be an ideal in $L$.
Let $x$ be an element of $S$.
Suppose $x \notin I$
Then there exists a prime ideal $P$ in $L$: $I \subseteq P$ and $x \notin P$ | By Upper Closure of Element is Filter:
:$x^\succeq$ is a filter.
We will prove that
:$I \cap x^\succeq = \O$
{{AimForCont}}:
:$\exists y: y \in I \cap x^\succeq$
By definition of intersection:
:$y \in I$ and $y \in x^\succeq$
By definition of upper closure of element:
:$x \preceq y$
By definition of lower section:
:$x ... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Distributive Lattice|distributive lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Let $x$ be an [[Definition:Element|element]] of $S$.
Suppose $x \notin I$
Then there exists a [[Definition:Prime Ideal (Order Theory)|prime ide... | By [[Upper Closure of Element is Filter]]:
:$x^\succeq$ is a [[Definition:Filter|filter]].
We will prove that
:$I \cap x^\succeq = \O$
{{AimForCont}}:
:$\exists y: y \in I \cap x^\succeq$
By definition of [[Definition:Set Intersection|intersection]]:
:$y \in I$ and $y \in x^\succeq$
By definition of [[Definition:Up... | If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element | https://proofwiki.org/wiki/If_Element_Does_Not_Belong_to_Ideal_then_There_Exists_Prime_Ideal_Including_Ideal_and_Excluding_Element | https://proofwiki.org/wiki/If_Element_Does_Not_Belong_to_Ideal_then_There_Exists_Prime_Ideal_Including_Ideal_and_Excluding_Element | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Distributive Lattice",
"Definition:Ideal (Order Theory)",
"Definition:Element",
"Definition:Prime Ideal (Order Theory)"
] | [
"Upper Closure of Element is Filter",
"Definition:Filter",
"Definition:Set Intersection",
"Definition:Upper Closure/Element",
"Definition:Lower Section",
"If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter",
"Definition:Prime Ideal (Order Theory)",
... |
proofwiki-12312 | Definite Integral to Infinity of Exponential of -x by Logarithm of x | Let $\ln t$ denote the natural logarithm function for real $t > 0$.
Let $e^{-t}$ denote the real exponential.
Then:
:$\ds \int_{0^+}^{\mathop \to +\infty} \ln t \, e^{-t} \rd t = - \gamma$
where the {{LHS}} is an improper integral, and $\gamma$ is the Euler-Mascheroni Constant. | {{begin-eqn}}
{{eqn | l = \int_{0^+}^{ \mathop \to +\infty} \ln t \, e^{-t} \rd t
| r = \int_{0^+}^{ \mathop \to +\infty} t^{1 - 1} \ln t \, e^{-t} \rd t
}}
{{eqn | r = \map {\Gamma'} 1
| c = Derivative of Gamma Function
}}
{{eqn | r = -\gamma
| c = Derivative of Gamma Function at $1$
}}
{{end-eqn}}
{... | Let $\ln t$ denote the [[Definition:Real Natural Logarithm|natural logarithm function]] for [[Definition:Real Number|real]] $t > 0$.
Let $e^{-t}$ denote the [[Definition:Real Exponential Function|real exponential]].
Then:
:$\ds \int_{0^+}^{\mathop \to +\infty} \ln t \, e^{-t} \rd t = - \gamma$
where the {{LHS}} is... | {{begin-eqn}}
{{eqn | l = \int_{0^+}^{ \mathop \to +\infty} \ln t \, e^{-t} \rd t
| r = \int_{0^+}^{ \mathop \to +\infty} t^{1 - 1} \ln t \, e^{-t} \rd t
}}
{{eqn | r = \map {\Gamma'} 1
| c = [[Derivative of Gamma Function]]
}}
{{eqn | r = -\gamma
| c = [[Derivative of Gamma Function at 1|Derivative o... | Definite Integral to Infinity of Exponential of -x by Logarithm of x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-x_by_Logarithm_of_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-x_by_Logarithm_of_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Exponential Function",
"Gamma Function"
] | [
"Definition:Natural Logarithm/Positive Real",
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Improper Integral",
"Definition:Euler-Mascheroni Constant"
] | [
"Derivative of Gamma Function",
"Derivative of Gamma Function at 1"
] |
proofwiki-12313 | Bottom not in Proper Filter | Let $L = \struct {S, \preceq}$ be a bounded below preordered set.
Let $F$ be a filter on $L$.
Then $F$ is proper filter {{iff}} $\bot \notin F$
where $\bot$ denotes the smallest element of $S$. | === Sufficient Condition ===
Suppose:
:$F$ is proper.
By definition of proper subset:
:$F \subseteq S$ and $F \ne S$
By definitions of set equality and subset:
:$\exists x: x \in S \land x \notin F$
By definition of smallest element:
:$\bot \preceq x$
Thus by definition of upper section:
:$\bot \notin F$
{{qed|lemma}} | Let $L = \struct {S, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Preordered Set|preordered set]].
Let $F$ be a [[Definition:Filter|filter]] on $L$.
Then $F$ is [[Definition:Proper Filter|proper filter]] {{iff}} $\bot \notin F$
where $\bot$ denotes the [[Definition:Smallest Element|sma... | === Sufficient Condition ===
Suppose:
:$F$ is [[Definition:Proper Subset|proper]].
By definition of [[Definition:Proper Subset|proper subset]]:
:$F \subseteq S$ and $F \ne S$
By definitions of [[Definition:Set Equality|set equality]] and [[Definition:Subset|subset]]:
:$\exists x: x \in S \land x \notin F$
By defini... | Bottom not in Proper Filter | https://proofwiki.org/wiki/Bottom_not_in_Proper_Filter | https://proofwiki.org/wiki/Bottom_not_in_Proper_Filter | [
"Preorder Theory"
] | [
"Definition:Bounded Below Set",
"Definition:Preordering/Preordered Set",
"Definition:Filter",
"Definition:Filter/Proper Filter",
"Definition:Smallest Element"
] | [
"Definition:Proper Subset",
"Definition:Proper Subset",
"Definition:Set Equality",
"Definition:Subset",
"Definition:Smallest Element",
"Definition:Upper Section",
"Definition:Set Equality",
"Definition:Proper Subset"
] |
proofwiki-12314 | Obtuse Triangle Divided into Acute Triangles | Let $T$ be an obtuse triangle.
Let $T$ be dissected into $n$ acute triangles.
Then $n \ge 7$. | As $D$ is equidistant from $AC$, $CB$ and $BA$, it follows that $\angle CDH = \angle CDE = \angle FDG$.
As $CD = DE = DF = DG = DH$, it follows that each of $\triangle CDE$, $\triangle CDH$ and $\triangle FDG$ are isosceles.
From Triangle Side-Angle-Side Congruence, $\triangle CDE$, $\triangle CDH$ and $\triangle FDG$ ... | Let $T$ be an [[Definition:Obtuse Triangle|obtuse triangle]].
Let $T$ be [[Definition:Dissection|dissected]] into $n$ [[Definition:Acute Triangle|acute triangles]].
Then $n \ge 7$. | As $D$ is equidistant from $AC$, $CB$ and $BA$, it follows that $\angle CDH = \angle CDE = \angle FDG$.
As $CD = DE = DF = DG = DH$, it follows that each of $\triangle CDE$, $\triangle CDH$ and $\triangle FDG$ are [[Definition:Isosceles Triangle|isosceles]].
From [[Triangle Side-Angle-Side Congruence]], $\triangle CD... | Obtuse Triangle Divided into Acute Triangles | https://proofwiki.org/wiki/Obtuse_Triangle_Divided_into_Acute_Triangles | https://proofwiki.org/wiki/Obtuse_Triangle_Divided_into_Acute_Triangles | [
"Recreational Mathematics",
"Obtuse Triangles",
"Acute Triangles",
"Dissections"
] | [
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Dissection",
"Definition:Triangle (Geometry)/Acute"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Triangle Side-Angle-Side Congruence",
"Definition:Congruence (Geometry)",
"Definition:Acute Angle",
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Right Angle",
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Acute Angle",
"Definiti... |
proofwiki-12315 | If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.
Let $I$ be an ideal in $L$.
Let $F$ be a filter on $L$ such that
:$F \cap I = \O$
Then there exists a prime filter $P$ in $L$:
$F \subseteq P$ and $P \cap I = \O$ | By Dual Distributive Lattice is Distributive:
:$L^{-1}$ is a distributive lattice
where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$.
By Filter is Ideal in Dual Ordered Set:
:$I' := F$ as an ideal in $L^{-1}$.
By Ideal is Filter in Dual Ordered Set:
:$F' := I$ as a filter on $L^{-1}$.
By assumption:
:$I' \ca... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Distributive Lattice|distributive lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Let $F$ be a [[Definition:Filter|filter]] on $L$ such that
:$F \cap I = \O$
Then there exists a [[Definition:Prime Filter (Order Theory)|prime f... | By [[Dual Distributive Lattice is Distributive]]:
:$L^{-1}$ is a [[Definition:Distributive Lattice|distributive lattice]]
where $L^{-1} = \struct {S, \succeq}$ denotes the [[Definition:Dual Ordered Set|dual]] of $L$.
By [[Filter is Ideal in Dual Ordered Set]]:
:$I' := F$ as an [[Definition:Ideal (Order Theory)|ideal]]... | If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal | https://proofwiki.org/wiki/If_Ideal_and_Filter_are_Disjoint_then_There_Exists_Prime_Filter_Including_Filter_and_Disjoint_from_Ideal | https://proofwiki.org/wiki/If_Ideal_and_Filter_are_Disjoint_then_There_Exists_Prime_Filter_Including_Filter_and_Disjoint_from_Ideal | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Distributive Lattice",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Prime Filter (Order Theory)"
] | [
"Dual Distributive Lattice is Distributive",
"Definition:Distributive Lattice",
"Definition:Dual Ordering/Dual Ordered Set",
"Filter is Ideal in Dual Ordered Set",
"Definition:Ideal (Order Theory)",
"Ideal is Filter in Dual Ordered Set",
"Definition:Filter",
"If Ideal and Filter are Disjoint then Ther... |
proofwiki-12316 | Dual Distributive Lattice is Distributive | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice.
Then
:$L$ is a distributive lattice
{{iff}}
:$L^{-1}$ is a distributive lattice
where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$. | === Sufficient Condition ===
Let $L$ be a distributive lattice.
By Dual of Lattice Ordering is Lattice Ordering:
:$L^{-1}$ is lattice.
Let $x, y, z \in S$.
$\vee'$ and $\wedge'$ denotes join and meet in $L^{-1}$.
Thus
{{begin-eqn}}
{{eqn | l = x \wedge' \paren {y \vee' z}
| r = x \wedge' \paren {y \wedge z}
... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Then
:$L$ is a [[Definition:Distributive Lattice|distributive lattice]]
{{iff}}
:$L^{-1}$ is a [[Definition:Distributive Lattice|distributive lattice]]
where $L^{-1} = \struct {S, \succeq}$ denotes the [[Definition:Dual Or... | === Sufficient Condition ===
Let $L$ be a [[Definition:Distributive Lattice|distributive lattice]].
By [[Dual of Lattice Ordering is Lattice Ordering]]:
:$L^{-1}$ is [[Definition:Lattice (Order Theory)|lattice]].
Let $x, y, z \in S$.
$\vee'$ and $\wedge'$ denotes [[Definition:Join (Order Theory)|join]] and [[Defini... | Dual Distributive Lattice is Distributive | https://proofwiki.org/wiki/Dual_Distributive_Lattice_is_Distributive | https://proofwiki.org/wiki/Dual_Distributive_Lattice_is_Distributive | [
"Distributive Lattices",
"Dual Orderings"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Distributive Lattice",
"Definition:Distributive Lattice",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Definition:Distributive Lattice",
"Dual of Lattice Ordering is Lattice Ordering",
"Definition:Lattice (Order Theory)",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Join is Dual to Meet",
"Join is Dual to Meet",
"Join is Dual to Meet",
"Join is Dual to Meet"
] |
proofwiki-12317 | Exchange of Rows as Sequence of Other Elementary Row Operations | Let $\mathbf A$ be an $m \times n$ matrix.
Let $i, j \in \closedint 1 m: i \ne j$
Let $r_k$ denote the $k$th row of $\mathbf A$ for $1 \le k \le m$:
:$r_k = \begin {pmatrix} a_{k 1} & a_{k 2} & \cdots & a_{k n} \end {pmatrix}$
Let $e$ be the elementary row operation acting on $\mathbf A$ as:
{{begin-axiom}}
{{axiom | n... | In the below:
:$r_i$ denotes the initial state of row $i$
:$r_j$ denotes the initial state of row $j$
:$r_i'$ denotes the state of row $i$ after having had the latest elementary row operation applied
:$r_j'$ denotes the state of row $j$ after having had the latest elementary row operation applied.
$(1)$: Apply $\text {... | Let $\mathbf A$ be an $m \times n$ [[Definition:Matrix|matrix]].
Let $i, j \in \closedint 1 m: i \ne j$
Let $r_k$ denote the $k$th [[Definition:Row of Matrix|row]] of $\mathbf A$ for $1 \le k \le m$:
:$r_k = \begin {pmatrix} a_{k 1} & a_{k 2} & \cdots & a_{k n} \end {pmatrix}$
Let $e$ be the [[Definition:Elementary... | In the below:
:$r_i$ denotes the initial state of [[Definition:Row of Matrix|row]] $i$
:$r_j$ denotes the initial state of [[Definition:Row of Matrix|row]] $j$
:$r_i'$ denotes the state of [[Definition:Row of Matrix|row]] $i$ after having had the latest [[Definition:Elementary Row Operation|elementary row operation]] ... | Exchange of Rows as Sequence of Other Elementary Row Operations | https://proofwiki.org/wiki/Exchange_of_Rows_as_Sequence_of_Other_Elementary_Row_Operations | https://proofwiki.org/wiki/Exchange_of_Rows_as_Sequence_of_Other_Elementary_Row_Operations | [
"Elementary Row Operations"
] | [
"Definition:Matrix",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row... | [
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Matri... |
proofwiki-12318 | Proper Filter is Included in Ultrafilter in Boolean Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a Boolean lattice.
Let $F$ be a proper filter on $L$.
Then there exists ultrafilter $G$ on $L$: $F \subseteq G$ | By Singleton of Bottom is Ideal:
:$I := \set \bot$ is an ideal in $L$.
where $\bot$ denotes the bottom of $L$.
We will prove that
:$F \cap I = \O$
Let $x \in I$.
By definition of singleton:
:$x = \bot$
Thus by Bottom not in Proper Filter:
:$x \notin F$
{{qed|lemma}}
By If Ideal and Filter are Disjoint then There Exists... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $F$ be a [[Definition:Proper Subset|proper]] [[Definition:Filter|filter]] on $L$.
Then there exists [[Definition:Ultrafilter (Order Theory)|ultrafilter]] $G$ on $L$: $F \subseteq G$ | By [[Singleton of Bottom is Ideal]]:
:$I := \set \bot$ is an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
where $\bot$ denotes the [[Definition:Bottom of Lattice|bottom]] of $L$.
We will prove that
:$F \cap I = \O$
Let $x \in I$.
By definition of [[Definition:Singleton|singleton]]:
:$x = \bot$
Thus by [[Bottom... | Proper Filter is Included in Ultrafilter in Boolean Lattice | https://proofwiki.org/wiki/Proper_Filter_is_Included_in_Ultrafilter_in_Boolean_Lattice | https://proofwiki.org/wiki/Proper_Filter_is_Included_in_Ultrafilter_in_Boolean_Lattice | [
"Order Theory"
] | [
"Definition:Boolean Lattice",
"Definition:Proper Subset",
"Definition:Filter",
"Definition:Ultrafilter (Order Theory)"
] | [
"Singleton of Bottom is Ideal",
"Definition:Ideal (Order Theory)",
"Definition:Bottom of Lattice",
"Definition:Singleton",
"Bottom not in Proper Filter",
"If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal",
"Definition:Prime Filter (Order Theory)",
... |
proofwiki-12319 | Seven Different Frieze Groups | There are $7$ different frieze groups. | {{ProofWanted|Lots of background work needed.}} | There are $7$ different [[Definition:Frieze Group|frieze groups]]. | {{ProofWanted|Lots of background work needed.}} | Seven Different Frieze Groups | https://proofwiki.org/wiki/Seven_Different_Frieze_Groups | https://proofwiki.org/wiki/Seven_Different_Frieze_Groups | [
"Frieze Groups",
"Symmetry Groups",
"7"
] | [
"Definition:Frieze Group"
] | [] |
proofwiki-12320 | Way Below iff Second Operand Preceding Supremum of Prime Ideal implies First Operand is Element of Ideal | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive complete lattice.
Let $x, y \in S$.
Then $x \ll y$ {{iff}}:
:for every prime ideal $P$ in $L$: $y \preceq \sup P \implies x \in P$ | === Sufficient Condition ===
The result follows by Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal.
{{qed|lemma}} | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Distributive Lattice|distributive]] [[Definition:Complete Lattice|complete lattice]].
Let $x, y \in S$.
Then $x \ll y$ {{iff}}:
:for every [[Definition:Prime Ideal (Order Theory)|prime ideal]] $P$ in $L$: $y \preceq \sup P \implies x \in P$ | === Sufficient Condition ===
The result follows by [[Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal]].
{{qed|lemma}} | Way Below iff Second Operand Preceding Supremum of Prime Ideal implies First Operand is Element of Ideal | https://proofwiki.org/wiki/Way_Below_iff_Second_Operand_Preceding_Supremum_of_Prime_Ideal_implies_First_Operand_is_Element_of_Ideal | https://proofwiki.org/wiki/Way_Below_iff_Second_Operand_Preceding_Supremum_of_Prime_Ideal_implies_First_Operand_is_Element_of_Ideal | [
"Prime Ideals (Order Theory)",
"Way Below Relation"
] | [
"Definition:Distributive Lattice",
"Definition:Complete Lattice",
"Definition:Prime Ideal (Order Theory)"
] | [
"Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal",
"Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal"
] |
proofwiki-12321 | Lower Closure is Prime Ideal for Prime Element | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice.
Let $p \in S$ be a prime element.
Then $p^\preceq$ is a prime ideal. | Let $x, y \in S$ such that
:$x \wedge y \in p^\preceq$
By definition of lower closure of element:
:$x \wedge y \preceq p$
By Characterization of Prime Ideal:
:$x \preceq p$ or $y \preceq p$
Thus by definition of lower closure of element:
:$x \in p^\preceq$ or $y \in p^\preceq$
{{qed}} | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $p \in S$ be a [[Definition:Prime Element (Order Theory)|prime element]].
Then $p^\preceq$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]]. | Let $x, y \in S$ such that
:$x \wedge y \in p^\preceq$
By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
:$x \wedge y \preceq p$
By [[Characterization of Prime Ideal]]:
:$x \preceq p$ or $y \preceq p$
Thus by definition of [[Definition:Lower Closure of Element|lower closure of elemen... | Lower Closure is Prime Ideal for Prime Element | https://proofwiki.org/wiki/Lower_Closure_is_Prime_Ideal_for_Prime_Element | https://proofwiki.org/wiki/Lower_Closure_is_Prime_Ideal_for_Prime_Element | [
"Prime Ideals (Order Theory)",
"Prime Elements"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Prime Ideal (Order Theory)"
] | [
"Definition:Lower Closure/Element",
"Characterization of Prime Ideal",
"Definition:Lower Closure/Element"
] |
proofwiki-12322 | Prime is Pseudoprime (Order Theory) | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice.
Let $p \in S$ be a prime element.
Then $p$ is pseudoprime. | By Lower Closure is Prime Ideal for Prime Element:
:$p^\preceq$ is prime ideal.
By Supremum of Lower Closure of Element:
:$ \sup \left({ p^\preceq }\right) = p$
Hence $p$ is pseudoprime.
{{qed}} | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Let $p \in S$ be a [[Definition:Prime Element (Order Theory)|prime element]].
Then $p$ is [[Definition:Pseudoprime (Order Theory)|pseudoprime]]. | By [[Lower Closure is Prime Ideal for Prime Element]]:
:$p^\preceq$ is [[Definition:Prime Ideal (Order Theory)|prime ideal]].
By [[Supremum of Lower Closure of Element]]:
:$ \sup \left({ p^\preceq }\right) = p$
Hence $p$ is [[Definition:Pseudoprime (Order Theory)|pseudoprime]].
{{qed}} | Prime is Pseudoprime (Order Theory) | https://proofwiki.org/wiki/Prime_is_Pseudoprime_(Order_Theory) | https://proofwiki.org/wiki/Prime_is_Pseudoprime_(Order_Theory) | [
"Prime Elements"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Pseudoprime (Order Theory)"
] | [
"Lower Closure is Prime Ideal for Prime Element",
"Definition:Prime Ideal (Order Theory)",
"Supremum of Lower Closure of Element",
"Definition:Pseudoprime (Order Theory)"
] |
proofwiki-12323 | Regular Heptagon is Smallest with no Compass and Straightedge Construction | The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction. | {{tidy}}
{{MissingLinks}}
A theorem states that:
"The n-gon is constructible by compass and straightedge construction if and only if $n = 2^kn_0$, with $k \in \mathbb{Z}_{\geq0}$ and $n_0$ the product of any number of distinct Fermat primes."
Note that 3 and 5 are both Fermat primes, so their respective n-gon are both ... | The [[Definition:Regular Heptagon|regular heptagon]] is the smallest [[Definition:Regular Polygon|regular polygon]] (smallest in the sense of having fewest [[Definition:Side of Polygon|sides]]) that cannot be constructed using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]]. | {{tidy}}
{{MissingLinks}}
A theorem states that:
"The n-gon is constructible by compass and straightedge construction if and only if $n = 2^kn_0$, with $k \in \mathbb{Z}_{\geq0}$ and $n_0$ the product of any number of distinct Fermat primes."
Note that 3 and 5 are both Fermat primes, so their respective n-gon are both... | Regular Heptagon is Smallest with no Compass and Straightedge Construction/Proof 2 | https://proofwiki.org/wiki/Regular_Heptagon_is_Smallest_with_no_Compass_and_Straightedge_Construction | https://proofwiki.org/wiki/Regular_Heptagon_is_Smallest_with_no_Compass_and_Straightedge_Construction/Proof_2 | [
"Regular Heptagon is Smallest with no Compass and Straightedge Construction",
"Compass and Straightedge Constructions",
"Regular Polygons",
"7"
] | [
"Definition:Heptagon/Regular",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Compass and Straightedge Construction"
] | [] |
proofwiki-12324 | Smallest Prime Number whose Period is of Maximum Length | $7$ is the smallest prime number the period of whose reciprocal, when expressed in decimal notation, is maximum:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ | From Maximum Period of Reciprocal of Prime, the maximum period of $\dfrac 1 p$ is $p - 1$.
:$\dfrac 1 2 = 0 \cdotp 5$: not recurring.
:$\dfrac 1 3 = 0 \cdotp \dot 3$: recurring with period $1$.
:$\dfrac 1 5 = 0 \cdotp 2$: not recurring.
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$: recurring with period $6$.
{{qed}} | $7$ is the smallest [[Definition:Prime Number|prime number]] the [[Definition:Period of Recurrence|period]] of whose [[Definition:Reciprocal|reciprocal]], when expressed in [[Definition:Decimal Notation|decimal notation]], is maximum:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ | From [[Maximum Period of Reciprocal of Prime]], the maximum [[Definition:Period of Recurrence|period]] of $\dfrac 1 p$ is $p - 1$.
:$\dfrac 1 2 = 0 \cdotp 5$: not recurring.
:$\dfrac 1 3 = 0 \cdotp \dot 3$: recurring with [[Definition:Period of Recurrence|period]] $1$.
:$\dfrac 1 5 = 0 \cdotp 2$: not recurring.
:$\... | Smallest Prime Number whose Period is of Maximum Length | https://proofwiki.org/wiki/Smallest_Prime_Number_whose_Period_is_of_Maximum_Length | https://proofwiki.org/wiki/Smallest_Prime_Number_whose_Period_is_of_Maximum_Length | [
"Fractions"
] | [
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Reciprocal",
"Definition:Decimal Notation"
] | [
"Maximum Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12325 | Characterization of Pseudoprime Element by Finite Infima | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a continuous lattice.
Let $p \in S$ be a pseudoprime element.
Let $A$ be a non-empty finite subset of $S$ such that
:$\inf A \ll p$
where $\ll$ denotes the way below relation.
Then $\exists a \in A: a \preceq p$ | By definition of pseudoprime element:
:there exists a prime ideal $P$ in $L$: $p = \sup P$
By definition of way below closure:
:$\inf A \in p^\ll$
By definition of reflexivity:
:$p \preceq \sup P$
By Continuous iff Way Below Closure is Ideal and Element Precedes Supremum:
:$p^\ll \subseteq P$
By definition of subset:
:... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Continuous Ordered Set|continuous]] [[Definition:Lattice (Order Theory)|lattice]].
Let $p \in S$ be a [[Definition:Pseudoprime (Order Theory)|pseudoprime element]].
Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Def... | By definition of [[Definition:Pseudoprime (Order Theory)|pseudoprime element]]:
:there exists a [[Definition:Prime Ideal (Order Theory)|prime ideal]] $P$ in $L$: $p = \sup P$
By definition of [[Definition:Way Below Closure|way below closure]]:
:$\inf A \in p^\ll$
By definition of [[Definition:Reflexivity|reflexivity]... | Characterization of Pseudoprime Element by Finite Infima | https://proofwiki.org/wiki/Characterization_of_Pseudoprime_Element_by_Finite_Infima | https://proofwiki.org/wiki/Characterization_of_Pseudoprime_Element_by_Finite_Infima | [
"Prime Elements"
] | [
"Definition:Continuous Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Pseudoprime (Order Theory)",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Element is Way Below"
] | [
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Way Below Closure",
"Definition:Reflexivity",
"Continuous iff Way Below Closure is Ideal and Element Precedes Supremum",
"Definition:Subset",
"Characterization of Prime Ideal by Finite Infima",
"Definition:Up... |
proofwiki-12326 | Elementary Row Operations as Matrix Multiplications/Corollary | Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary row operation.
Then there exists an elementary row matrix of order $m$ such that:
:$\mathbf {E X} = \mathbf Y$ | Let $e$ be the elementary row operation such that $e \paren {\mathbf X} = \mathbf Y$.
Then this result follows immediately from Elementary Row Operations as Matrix Multiplications:
:$e \paren {\mathbf X} = \mathbf {E X} = \mathbf Y$
where $\mathbf E = e \paren {\mathbf I}$.
{{qed}} | Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ [[Definition:Matrix|matrices]] that differ by exactly one [[Definition:Elementary Row Operation|elementary row operation]].
Then there exists an [[Definition:Elementary Row Matrix|elementary row matrix]] of [[Definition:Order of Square Matrix|order]] $m$ such that:
... | Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] such that $e \paren {\mathbf X} = \mathbf Y$.
Then this result follows immediately from [[Elementary Row Operations as Matrix Multiplications]]:
:$e \paren {\mathbf X} = \mathbf {E X} = \mathbf Y$
where $\mathbf E = e \paren {\mathbf I}$... | Elementary Row Operations as Matrix Multiplications/Corollary | https://proofwiki.org/wiki/Elementary_Row_Operations_as_Matrix_Multiplications/Corollary | https://proofwiki.org/wiki/Elementary_Row_Operations_as_Matrix_Multiplications/Corollary | [
"Conventional Matrix Multiplication",
"Elementary Row Operations"
] | [
"Definition:Matrix",
"Definition:Elementary Operation/Row",
"Definition:Elementary Matrix/Row Operation",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Elementary Operation/Row",
"Elementary Row Operations as Matrix Multiplications"
] |
proofwiki-12327 | Cyclotomic Polynomial of Prime Index | Let $p$ be a prime number.
The '''$p$th cyclotomic polynomial''' is:
:$\map {\Phi_p} x = x^{p - 1} + x^{p - 2} + \cdots + x + 1$ | From Product of Cyclotomic Polynomials:
:$\map {\Phi_p} x \map {\Phi_1} x = x^p - 1$
Thus from Sum of Geometric Sequence:
:$\map {\Phi_p} x = \dfrac {x^p - 1} {x - 1} = x^{p - 1} + x^{p - 2} + \cdots + x + 1$
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
The '''$p$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]]''' is:
:$\map {\Phi_p} x = x^{p - 1} + x^{p - 2} + \cdots + x + 1$ | From [[Product of Cyclotomic Polynomials]]:
:$\map {\Phi_p} x \map {\Phi_1} x = x^p - 1$
Thus from [[Sum of Geometric Sequence]]:
:$\map {\Phi_p} x = \dfrac {x^p - 1} {x - 1} = x^{p - 1} + x^{p - 2} + \cdots + x + 1$
{{qed}} | Cyclotomic Polynomial of Prime Index | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Prime_Index | https://proofwiki.org/wiki/Cyclotomic_Polynomial_of_Prime_Index | [
"Examples of Cyclotomic Polynomials"
] | [
"Definition:Prime Number",
"Definition:Cyclotomic Polynomial"
] | [
"Product of Cyclotomic Polynomials",
"Sum of Geometric Sequence"
] |
proofwiki-12328 | Product of Cyclotomic Polynomials | Let $n > 0$ be a (strictly) positive integer.
Then:
:$\ds \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$
where:
:$\map {\Phi_d} x$ denotes the $d$th cyclotomic polynomial
:the product runs over all divisors of $n$. | From the Polynomial Factor Theorem and Complex Roots of Unity in Exponential Form:
:$\ds x^n - 1 = \prod_\zeta \paren {x - \zeta}$
where the product runs over all complex $n$th roots of unity.
In the {{LHS}}, each factor $x - \zeta$ appears exactly once, in the factorization of $\map {\Phi_d} x$ where $d$ is the order ... | Let $n > 0$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$
where:
:$\map {\Phi_d} x$ denotes the $d$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]]
:the product runs over all [[Definition:Divisor of Integer|... | From the [[Polynomial Factor Theorem]] and [[Complex Roots of Unity in Exponential Form]]:
:$\ds x^n - 1 = \prod_\zeta \paren {x - \zeta}$
where the product runs over all [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
In the {{LHS}}, each [[Definition:Factor of Polynomial|factor]] $x - \zeta$ app... | Product of Cyclotomic Polynomials | https://proofwiki.org/wiki/Product_of_Cyclotomic_Polynomials | https://proofwiki.org/wiki/Product_of_Cyclotomic_Polynomials | [
"Cyclotomic Polynomials"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Cyclotomic Polynomial",
"Definition:Divisor (Algebra)/Integer"
] | [
"Polynomial Factor Theorem",
"Complex Roots of Unity in Exponential Form",
"Definition:Root of Unity/Complex",
"Definition:Divisor of Polynomial",
"Definition:Root of Unity/Order"
] |
proofwiki-12329 | First Cyclotomic Polynomial | The '''first cyclotomic polynomial''' is:
:$\map {\Phi_1} x = x - 1$ | By definition:
:$\ds \map {\Phi_1} x = \prod_\zeta \paren {x - \zeta}$
where the product runs over all primitive complex first roots of unity.
A root of unity has order $1$ {{iff}} it equals $1$.
{{explain|The above statement needs justification, with reference to the definition of the order of a root of unity.}}
Hence... | The '''first [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]]''' is:
:$\map {\Phi_1} x = x - 1$ | By definition:
:$\ds \map {\Phi_1} x = \prod_\zeta \paren {x - \zeta}$
where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex first roots of unity]].
A [[Definition:Root of Unity|root of unity]] has order $1$ {{iff}} it equals $1$.
{{explain|The above statement needs justificat... | First Cyclotomic Polynomial | https://proofwiki.org/wiki/First_Cyclotomic_Polynomial | https://proofwiki.org/wiki/First_Cyclotomic_Polynomial | [
"Examples of Cyclotomic Polynomials"
] | [
"Definition:Cyclotomic Polynomial"
] | [
"Definition:Root of Unity/Complex/Primitive",
"Definition:Root of Unity",
"Category:Examples of Cyclotomic Polynomials"
] |
proofwiki-12330 | Cyclotomic Polynomial has Integer Coefficients | Let $n \in \Z_{>0}$ be a positive integer.
Then the $n$th cyclotomic polynomial $\map {\Phi_n} x$ has integer coefficients. | The proof proceeds by strong induction on $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\map {\Phi_n} x$ has integer coefficients | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Then the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]] $\map {\Phi_n} x$ has [[Definition:Integer|integer]] [[Definition:Coefficient of Polynomial|coefficients]]. | The proof proceeds by [[Principle of Strong Induction|strong induction]] on $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {\Phi_n} x$ has [[Definition:Integer|integer]] [[Definition:Coefficient of Polynomial|coefficients]] | Cyclotomic Polynomial has Integer Coefficients | https://proofwiki.org/wiki/Cyclotomic_Polynomial_has_Integer_Coefficients | https://proofwiki.org/wiki/Cyclotomic_Polynomial_has_Integer_Coefficients | [
"Cyclotomic Polynomials"
] | [
"Definition:Positive/Integer",
"Definition:Cyclotomic Polynomial",
"Definition:Integer",
"Definition:Coefficient of Polynomial"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Integer",
"Definition:Coefficient of Polynomial",
"Definition:Integer",
"Definition:Coefficient of Polynomial",
"Definition:Integer",
"Definition:Coefficient of Polynomial",
"Definition:Integer",
"Definition:Coeffi... |
proofwiki-12331 | Formal Derivative of Polynomials Satisfies Leibniz's Rule | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be the polynomial ring over $R$.
Let $f, g \in R \sqbrk X$ be polynomials.
Let $f'$ and $g'$ denote their formal derivatives.
Then:
:$\paren {f g}' = f g' + f' g$ | Both sides are bilinear functions of $f$ and $g$, so it suffices to verify the equality in the case where $\map f X = X^n$ and $\map g x = X^m$.
Then:
:$\paren {X^n X^m}' = \paren {n + m} X^{n + m - 1}$
and:
:$\paren {X^n}' X^m + X^n \paren {X^m}' = n X^{n - 1} X^m + m X^n X^{m - 1}$
{{qed}}
Category:Leibniz's Rule
Cat... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $R \sqbrk X$ be the [[Definition:Polynomial Ring|polynomial ring]] over $R$.
Let $f, g \in R \sqbrk X$ be [[Definition:Polynomial (Abstract Algebra)|polynomials]].
Let $f'$ and $g'$ denote their [[Definition:Formal Derivative o... | Both sides are [[Definition:Bilinear Mapping|bilinear functions]] of $f$ and $g$, so it suffices to verify the equality in the case where $\map f X = X^n$ and $\map g x = X^m$.
Then:
:$\paren {X^n X^m}' = \paren {n + m} X^{n + m - 1}$
and:
:$\paren {X^n}' X^m + X^n \paren {X^m}' = n X^{n - 1} X^m + m X^n X^{m - 1}$
{... | Formal Derivative of Polynomials Satisfies Leibniz's Rule | https://proofwiki.org/wiki/Formal_Derivative_of_Polynomials_Satisfies_Leibniz's_Rule | https://proofwiki.org/wiki/Formal_Derivative_of_Polynomials_Satisfies_Leibniz's_Rule | [
"Leibniz's Rule",
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial over Ring",
"Definition:Formal Derivative of Polynomial"
] | [
"Definition:Bilinear Mapping",
"Category:Leibniz's Rule",
"Category:Polynomial Theory"
] |
proofwiki-12332 | Double Root of Polynomial is Root of Derivative | Let $R$ be a commutative ring with unity.
Let $f \in R \sqbrk X$ be a polynomial.
Let $a \in R$ be a root of $f$ with multiplicity at least $2$.
Let $f'$ denote the formal derivative of $f$.
Then $a$ is a root of $f'$. | Because $a$ has multiplicity at least $2$, we can write:
:$\map f X = \paren {X - a}^2 \map g X$
with $\map g X \in R \sqbrk X$.
From Formal Derivative of Polynomials Satisfies Leibniz's Rule:
:$\map {f'} X = 2 \paren {X - a} \map g X + \paren {X - a}^2 \map {g'} X$
and thus:
:$\map {f'} a = 0$
{{qed}}
Category:Polynom... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $f \in R \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]].
Let $a \in R$ be a [[Definition:Root of Polynomial|root]] of $f$ with [[Definition:Multiplicity (Polynomial)|multiplicity]] at least $2$.
Let $f'$... | Because $a$ has [[Definition:Multiplicity (Polynomial)|multiplicity]] at least $2$, we can write:
:$\map f X = \paren {X - a}^2 \map g X$
with $\map g X \in R \sqbrk X$.
From [[Formal Derivative of Polynomials Satisfies Leibniz's Rule]]:
:$\map {f'} X = 2 \paren {X - a} \map g X + \paren {X - a}^2 \map {g'} X$
and ... | Double Root of Polynomial is Root of Derivative | https://proofwiki.org/wiki/Double_Root_of_Polynomial_is_Root_of_Derivative | https://proofwiki.org/wiki/Double_Root_of_Polynomial_is_Root_of_Derivative | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial over Ring",
"Definition:Root of Polynomial",
"Definition:Multiplicity (Polynomial)",
"Definition:Formal Derivative of Polynomial",
"Definition:Root of Polynomial"
] | [
"Definition:Multiplicity (Polynomial)",
"Formal Derivative of Polynomials Satisfies Leibniz's Rule",
"Category:Polynomial Theory"
] |
proofwiki-12333 | 7 Prime Knots with 7 Crossings | There exist exactly $7$ prime knots which have exactly $7$ crossings. | {{ProofWanted|Much background work to be done.}} | There exist exactly $7$ [[Definition:Prime Knot|prime knots]] which have exactly $7$ [[Definition:Crossing|crossings]]. | {{ProofWanted|Much background work to be done.}} | 7 Prime Knots with 7 Crossings | https://proofwiki.org/wiki/7_Prime_Knots_with_7_Crossings | https://proofwiki.org/wiki/7_Prime_Knots_with_7_Crossings | [
"Knot Theory",
"7"
] | [
"Definition:Prime Knot",
"Definition:Crossing"
] | [] |
proofwiki-12334 | Prime Divisors of Cyclotomic Polynomials | Let $n \ge 1$ be a positive integer.
Let $\map {\Phi_n} x$ denote the $n$th cyclotomic polynomial.
Let $a \in \Z$ be an integer such that $\map {\Phi_n} a \ne 0$.
Let $p$ be a prime divisor of $\map {\Phi_n} a$.
Then $p \equiv 1 \pmod n$ or $p \divides n$. | Let $k$ be the order of $a$ modulo $p$.
By Element to Power of Multiple of Order is Identity, $k \divides p - 1$.
If $k = n$, the result follows.
Let $k < n$.
Then by Product of Cyclotomic Polynomials, there exists $d \divides k$ such that $p \divides \map {\Phi_d} a$.
Consequently, $a$ is a double root of $\Phi_d \Phi... | Let $n \ge 1$ be a [[Definition:Positive Integer|positive integer]].
Let $\map {\Phi_n} x$ denote the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
Let $a \in \Z$ be an [[Definition:Integer|integer]] such that $\map {\Phi_n} a \ne 0$.
Let $p$ be a [[Definition:Prime Divisor|prime divisor]] of $\m... | Let $k$ be the order of $a$ modulo $p$.
By [[Element to Power of Multiple of Order is Identity]], $k \divides p - 1$.
If $k = n$, the result follows.
Let $k < n$.
Then by [[Product of Cyclotomic Polynomials]], there exists $d \divides k$ such that $p \divides \map {\Phi_d} a$.
Consequently, $a$ is a [[Definition:... | Prime Divisors of Cyclotomic Polynomials | https://proofwiki.org/wiki/Prime_Divisors_of_Cyclotomic_Polynomials | https://proofwiki.org/wiki/Prime_Divisors_of_Cyclotomic_Polynomials | [
"Number Theory",
"Cyclotomic Polynomials"
] | [
"Definition:Positive/Integer",
"Definition:Cyclotomic Polynomial",
"Definition:Integer",
"Definition:Prime Factor"
] | [
"Element to Power of Multiple of Order is Identity",
"Product of Cyclotomic Polynomials",
"Definition:Multiplicity (Polynomial)",
"Product of Cyclotomic Polynomials",
"Double Root of Polynomial is Root of Derivative"
] |
proofwiki-12335 | Pack of Cards is Randomized by 7 Riffle Shuffles | A pack of cards is randomized by $7$ riffle shuffles. | {{ProofWanted|Background work needed.}} | A [[Definition:Pack of Cards|pack of cards]] is randomized by $7$ [[Definition:Riffle Shuffle|riffle shuffles]]. | {{ProofWanted|Background work needed.}} | Pack of Cards is Randomized by 7 Riffle Shuffles | https://proofwiki.org/wiki/Pack_of_Cards_is_Randomized_by_7_Riffle_Shuffles | https://proofwiki.org/wiki/Pack_of_Cards_is_Randomized_by_7_Riffle_Shuffles | [
"Combinatorics"
] | [
"Definition:Pack of Cards",
"Definition:Riffle Shuffle"
] | [] |
proofwiki-12336 | Sum of Complex Indices of Real Number | Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Let $\psi, \tau \in \C$ be complex numbers.
Let $r^\lambda$ be defined as the the principal branch of a positive real number raised to a complex number.
Then:
:$r^{\psi \mathop + \tau} = r^\psi \times r^\tau$ | Then:
{{begin-eqn}}
{{eqn | l = r^{\psi \mathop + \tau}
| r = \map \exp {\paren {\psi + \tau} \ln r}
| c = {{Defof|Power (Algebra)/Complex Number/Principal Branch/Positive Real Base|Principal Branch of Positive Real Number raised to Complex Number}}
}}
{{eqn | r = \map \exp {\psi \ln r + \tau \ln r}
}}
{{eq... | Let $r \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $\psi, \tau \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $r^\lambda$ be defined as the [[Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base|the principal branch of a ... | Then:
{{begin-eqn}}
{{eqn | l = r^{\psi \mathop + \tau}
| r = \map \exp {\paren {\psi + \tau} \ln r}
| c = {{Defof|Power (Algebra)/Complex Number/Principal Branch/Positive Real Base|Principal Branch of Positive Real Number raised to Complex Number}}
}}
{{eqn | r = \map \exp {\psi \ln r + \tau \ln r}
}}
{{eq... | Sum of Complex Indices of Real Number | https://proofwiki.org/wiki/Sum_of_Complex_Indices_of_Real_Number | https://proofwiki.org/wiki/Sum_of_Complex_Indices_of_Real_Number | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Complex Number",
"Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base"
] | [
"Exponential of Sum/Complex Numbers",
"Category:Powers"
] |
proofwiki-12337 | Universal Property of Free Modules | Let $R$ be a ring.
Let $M$ be a free $R$-module with basis $\set {e_i: i\in I}$.
Let $N$ be an $R$-module.
Let $\set {n_i: i \in I}$ be a family of elements of $N$.
Then there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\in I$. | Combine Free Module is Isomorphic to Free Module Indexed by Set and Universal Property of Free Module on Set.
{{handwaving}}
Category:Free Modules
Category:Module Theory
Category:Universal Properties
5dah6s5fykqiqimrzkusypeeibpbmzw | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] with [[Definition:Basis (Linear Algebra)|basis]] $\set {e_i: i\in I}$.
Let $N$ be an [[Definition:Module over Ring|$R$-module]].
Let $\set {n_i: i \in I}$ be a family of elements of $N$.
Then... | Combine [[Free Module is Isomorphic to Free Module Indexed by Set]] and [[Universal Property of Free Module on Set]].
{{handwaving}}
[[Category:Free Modules]]
[[Category:Module Theory]]
[[Category:Universal Properties]]
5dah6s5fykqiqimrzkusypeeibpbmzw | Universal Property of Free Modules | https://proofwiki.org/wiki/Universal_Property_of_Free_Modules | https://proofwiki.org/wiki/Universal_Property_of_Free_Modules | [
"Free Modules",
"Module Theory",
"Universal Properties"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Free Module over Ring",
"Definition:Basis (Linear Algebra)",
"Definition:Module over Ring"
] | [
"Free Module is Isomorphic to Free Module on Set",
"Universal Property of Free Module on Set",
"Category:Free Modules",
"Category:Module Theory",
"Category:Universal Properties"
] |
proofwiki-12338 | Odd Squares 7 Less than Nearest Power of 2 | There exist exactly $3$ odd squares which are $7$ less than the nearest power of $2$:
:$5^2 = 25 = 2^5 - 7$
:$11^2 = 121 = 2^7 - 7$
:$181^2 = 32 \, 761 = 2^{15} - 7$
{{OEIS|A038198}}
Note that this sequence includes $1$ and $3$, being all the squares which are $7$ less than a power of $2$.
However, for $1$ and $3$, tho... | From Solutions of Ramanujan-Nagell Equation, the only solutions to the equation:
:$x^2 + 7 = 2^n$
are $\tuple {x, n} =$:
:$\tuple {1, 3}, \tuple {3, 4}, \tuple {5, 5}, \tuple {11, 7}, \tuple {181, 15}$
so no more solutions exist.
{{qed}} | There exist exactly $3$ [[Definition:Odd Integer|odd]] [[Definition:Square Number|squares]] which are $7$ less than the nearest [[Definition:Integer Power|power]] of $2$:
:$5^2 = 25 = 2^5 - 7$
:$11^2 = 121 = 2^7 - 7$
:$181^2 = 32 \, 761 = 2^{15} - 7$
{{OEIS|A038198}}
Note that this sequence includes $1$ and $3$, being... | From [[Solutions of Ramanujan-Nagell Equation]], the only solutions to the equation:
:$x^2 + 7 = 2^n$
are $\tuple {x, n} =$:
:$\tuple {1, 3}, \tuple {3, 4}, \tuple {5, 5}, \tuple {11, 7}, \tuple {181, 15}$
so no more solutions exist.
{{qed}} | Odd Squares 7 Less than Nearest Power of 2 | https://proofwiki.org/wiki/Odd_Squares_7_Less_than_Nearest_Power_of_2 | https://proofwiki.org/wiki/Odd_Squares_7_Less_than_Nearest_Power_of_2 | [
"Number Theory"
] | [
"Definition:Odd Integer",
"Definition:Square Number",
"Definition:Power (Algebra)/Integer",
"Definition:Square Number",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Solutions of Ramanujan-Nagell Equation"
] |
proofwiki-12339 | Successive Solutions of Phi of n equals Phi of n + 2 | $7$ and $8$ are two successive integers which are solutions to the equation:
:$\map \phi n = \map \phi {n + 2}$ | From Euler Phi Function of Prime:
:$\map \phi 7 = 7 - 1 = 6$
From Euler Phi Function of Prime Power:
:$\map \phi 9 = \map \phi {3^2} = 2 \times 3^{2 - 1} = 6 = \map \phi 7$
From {{Corollary|Euler Phi Function of Prime Power}}:
:$\map \phi 8 = \map \phi {2^3} = 2^{3 - 1} = 4$
From Euler Phi Function of Integer:
:$\map \... | $7$ and $8$ are two successive [[Definition:Integer|integers]] which are solutions to the equation:
:$\map \phi n = \map \phi {n + 2}$ | From [[Euler Phi Function of Prime]]:
:$\map \phi 7 = 7 - 1 = 6$
From [[Euler Phi Function of Prime Power]]:
:$\map \phi 9 = \map \phi {3^2} = 2 \times 3^{2 - 1} = 6 = \map \phi 7$
From {{Corollary|Euler Phi Function of Prime Power}}:
:$\map \phi 8 = \map \phi {2^3} = 2^{3 - 1} = 4$
From [[Euler Phi Function of Inte... | Successive Solutions of Phi of n equals Phi of n + 2 | https://proofwiki.org/wiki/Successive_Solutions_of_Phi_of_n_equals_Phi_of_n_+_2 | https://proofwiki.org/wiki/Successive_Solutions_of_Phi_of_n_equals_Phi_of_n_+_2 | [
"Euler Phi Function"
] | [
"Definition:Integer"
] | [
"Euler Phi Function of Prime",
"Euler Phi Function of Prime Power",
"Euler Phi Function of Integer"
] |
proofwiki-12340 | Sums of Sequences of Consecutive Squares which are Square | The sums of the following sequences of successive squares are themselves square:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 7}^{29} k^2
| r = 7^2 + 8^2 + \cdots + 29^2
| c =
}}
{{eqn | l = \sum_{i \mathop = 7}^{39} k^2
| r = 7^2 + 8^2 + \cdots + 39^2
| c =
}}
{{eqn | l = \sum_{i \mathop = 7}^... | From Sum of Sequence of Squares:
{{:Sum of Sequence of Squares}}
Thus:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 7}^{29} i^2
| r = \sum_{i \mathop = 1}^{29} i^2 - \sum_{i \mathop = 1}^6 i^2
| c =
}}
{{eqn | r = \frac {29 \left({29 + 1}\right) \left({2 \times 29 + 1}\right)} 6 - \frac {6 \left({6 + 1}\rig... | The sums of the following [[Definition:Finite Sequence|sequences]] of successive [[Definition:Square Number|squares]] are themselves [[Definition:Square Number|square]]:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 7}^{29} k^2
| r = 7^2 + 8^2 + \cdots + 29^2
| c =
}}
{{eqn | l = \sum_{i \mathop = 7}^{39} k... | From [[Sum of Sequence of Squares]]:
{{:Sum of Sequence of Squares}}
Thus:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 7}^{29} i^2
| r = \sum_{i \mathop = 1}^{29} i^2 - \sum_{i \mathop = 1}^6 i^2
| c =
}}
{{eqn | r = \frac {29 \left({29 + 1}\right) \left({2 \times 29 + 1}\right)} 6 - \frac {6 \left({6 + 1... | Sums of Sequences of Consecutive Squares which are Square | https://proofwiki.org/wiki/Sums_of_Sequences_of_Consecutive_Squares_which_are_Square | https://proofwiki.org/wiki/Sums_of_Sequences_of_Consecutive_Squares_which_are_Square | [
"Square Numbers",
"Sums of Sequences"
] | [
"Definition:Finite Sequence",
"Definition:Square Number",
"Definition:Square Number"
] | [
"Sum of Sequence of Squares"
] |
proofwiki-12341 | Morphism from Ring with Unity to Module | Let $R$ be a ring with unity.
Let $M$ be an $R$-module.
Then for every $m \in M$ there exists a unique $R$-module morphism:
:$\psi: R \to M$
that sends $1$ to $m$. | === Existence ===
Let $r \in R$.
Let $\map \psi r := r m$.
This map is $R$-linear by definition of a module.
{{qed|lemma}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be an [[Definition:Module over Ring|$R$-module]].
Then for every $m \in M$ there exists a unique $R$-module morphism:
:$\psi: R \to M$
that sends $1$ to $m$. | === Existence ===
Let $r \in R$.
Let $\map \psi r := r m$.
This map is $R$-linear by definition of a [[Definition:Module over Ring|module]].
{{qed|lemma}} | Morphism from Ring with Unity to Module | https://proofwiki.org/wiki/Morphism_from_Ring_with_Unity_to_Module | https://proofwiki.org/wiki/Morphism_from_Ring_with_Unity_to_Module | [
"Rings with Unity",
"Module Theory"
] | [
"Definition:Ring with Unity",
"Definition:Module over Ring"
] | [
"Definition:Module over Ring"
] |
proofwiki-12342 | Universal Property of Free Module on Set | Let $R$ be a ring with unity.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Let $M$ be an $R$-module.
Let $\family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Then there exists a unique $R$-module morphism:
:$\Psi: R^{\paren I} \to M$
that sends the $i$th canonical basis element to $m_i$, for all $i \i... | === Existence ===
By Morphism from Ring with Unity to Module, for all $i$ there exists a morphism:
:$\psi_i: R \to M$
with $\map {\psi_i} 1 = m_i$.
By Universal Property of Direct Sum of Modules, there exists a morphism:
:$\Psi: R^{\paren I} \to M$
such that $\Psi \circ \iota_i = \psi_i$ for all $i$.
Thus:
:$\forall i ... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $R^{\paren I}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
Let $M$ be an [[Definition:Module over Ring|$R$-module]].
Let $\family {m_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of elements of $M$.
Then there e... | === Existence ===
By [[Morphism from Ring with Unity to Module]], for all $i$ there exists a morphism:
:$\psi_i: R \to M$
with $\map {\psi_i} 1 = m_i$.
By [[Universal Property of Direct Sum of Modules]], there exists a morphism:
:$\Psi: R^{\paren I} \to M$
such that $\Psi \circ \iota_i = \psi_i$ for all $i$.
Thu... | Universal Property of Free Module on Set | https://proofwiki.org/wiki/Universal_Property_of_Free_Module_on_Set | https://proofwiki.org/wiki/Universal_Property_of_Free_Module_on_Set | [
"Module Theory",
"Direct Products",
"Universal Properties"
] | [
"Definition:Ring with Unity",
"Definition:Free Module on Set",
"Definition:Module over Ring",
"Definition:Indexing Set/Family",
"Definition:R-Algebraic Structure Homomorphism",
"Definition:Canonical Basis of Free Module on Set"
] | [
"Morphism from Ring with Unity to Module",
"Universal Property of Direct Sum of Modules",
"Morphism from Ring with Unity to Module",
"Universal Property of Direct Sum of Modules"
] |
proofwiki-12343 | Multiplicative Auxiliary Relation iff Images are Filtered | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice.
Let $\RR$ be an auxiliary relation on $S$.
Then $\RR$ is multiplicative {{iff}}:
:for all $x \in S$: $\map \RR x$ is filtered
where $\map \RR x$ denotes the $\RR$-image of $x$. | === Sufficient Condition ===
Let $\RR$ be multiplicative.
Let $x \in S$.
Let $a, b \in \map \RR x$.
By definition of $\RR$-image of element:
:$\tuple {x, a}, \tuple {x, b} \in \RR$
By definition of multiplicative relation:
:$\tuple {x, a \wedge b} \in \RR$
By definition of $\RR$-image of element:
:$a \wedge b \in \map ... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Lattice (Order Theory)|lattice]].
Let $\RR$ be an [[Definition:Auxiliary Relation|auxiliary relation]] on $S$.
Then $\RR$ is [[Definition:Multiplicative Relation|multiplicative]] {{iff}}:
:for all $x \in S$:... | === Sufficient Condition ===
Let $\RR$ be [[Definition:Multiplicative Relation|multiplicative]].
Let $x \in S$.
Let $a, b \in \map \RR x$.
By definition of [[Definition:Image of Element under Relation|$\RR$-image of element]]:
:$\tuple {x, a}, \tuple {x, b} \in \RR$
By definition of [[Definition:Multiplicative Rel... | Multiplicative Auxiliary Relation iff Images are Filtered | https://proofwiki.org/wiki/Multiplicative_Auxiliary_Relation_iff_Images_are_Filtered | https://proofwiki.org/wiki/Multiplicative_Auxiliary_Relation_iff_Images_are_Filtered | [
"Order Theory",
"Auxiliary Relations"
] | [
"Definition:Bounded Below Set",
"Definition:Lattice (Order Theory)",
"Definition:Auxiliary Relation",
"Definition:Multiplicative Relation",
"Definition:Filtered Subset",
"Definition:Image (Set Theory)/Relation/Element"
] | [
"Definition:Multiplicative Relation",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Multiplicative Relation",
"Definition:Image (Set Theory)/Relation/Element",
"Meet Precedes Operands",
"Definition:Filtered Subset",
"Definition:Filtered Subset",
"Definition:Image (Set Theory)/Relation/... |
proofwiki-12344 | Characterisation of Spanning Set through Free Module Indexed by Set | Let $M$ be a unitary $R$-module.
Let $S = \family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by Universal Property of Free Module Indexed by Set.
Then $S$ is a spanning set of $M$ {{iff}} $\Psi$ is surjective. | {{MissingLinks|linear combination}}
For $\family {r_i}_{i \mathop \in I} \in R^{\paren I}$ we have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = \ds \sum_{i \mathop \in I} m_i r_i$
Thus $\Psi$ is surjective {{iff}} every element of $M$ is a linear combination of $S$.
{{qed}}
Category:Module Theory
htzgah8boo1b4rbkl... | Let $M$ be a [[Definition:Unitary Module|unitary $R$-module]].
Let $S = \family {m_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Element|elements]] of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by [[Universal Property of Free Module Indexed by Set]].
Then $S$ is a [[... | {{MissingLinks|linear combination}}
For $\family {r_i}_{i \mathop \in I} \in R^{\paren I}$ we have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = \ds \sum_{i \mathop \in I} m_i r_i$
Thus $\Psi$ is [[Definition:Surjection|surjective]] {{iff}} every [[Definition:Element|element]] of $M$ is a linear combination of $S... | Characterisation of Spanning Set through Free Module Indexed by Set | https://proofwiki.org/wiki/Characterisation_of_Spanning_Set_through_Free_Module_Indexed_by_Set | https://proofwiki.org/wiki/Characterisation_of_Spanning_Set_through_Free_Module_Indexed_by_Set | [
"Module Theory"
] | [
"Definition:Unitary Module over Ring",
"Definition:Indexing Set/Family",
"Definition:Element",
"Universal Property of Free Module on Set",
"Definition:Generator of Module",
"Definition:Surjection"
] | [
"Definition:Surjection",
"Definition:Element",
"Category:Module Theory"
] |
proofwiki-12345 | Characterisation of Linearly Independent Set through Free Module Indexed by Set | Let $M$ be a unitary $R$-module.
Let $S = \family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi : R^{\paren I} \to M$ be the module homomorphism given by Universal Property of Free Module on Set.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$S$ linearly independent}}
{{item|(2):|$\Psi$ is injective}}
{{e... | We have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = 0$
{{iff}}:
:$\ds \sum_{i \mathop \in I} r_i m_i = 0$
{{explain|Justify the above statement.}}
Thus injectivity and linearly independent are equivalent.
{{explain|Justify the above statement.}}
{{qed}}
Category:Module Theory
fzdwqw7uhtlv1j4xuiti9isb1b9oab9 | Let $M$ be a [[Definition:Unitary Module|unitary $R$-module]].
Let $S = \family {m_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Element|elements]] of $M$.
Let $\Psi : R^{\paren I} \to M$ be the [[Definition:Module Homomorphism|module homomorphism]] given by [[Universal Property of F... | We have:
:$\map \Psi {\family {r_i}_{i \mathop \in I} } = 0$
{{iff}}:
:$\ds \sum_{i \mathop \in I} r_i m_i = 0$
{{explain|Justify the above statement.}}
Thus [[Definition:Injection|injectivity]] and [[Definition:Linearly Independent|linearly independent]] are equivalent.
{{explain|Justify the above statement.}}
{{q... | Characterisation of Linearly Independent Set through Free Module Indexed by Set | https://proofwiki.org/wiki/Characterisation_of_Linearly_Independent_Set_through_Free_Module_Indexed_by_Set | https://proofwiki.org/wiki/Characterisation_of_Linearly_Independent_Set_through_Free_Module_Indexed_by_Set | [
"Module Theory"
] | [
"Definition:Unitary Module over Ring",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Linear Transformation",
"Universal Property of Free Module on Set",
"Definition:Linearly Independent",
"Definition:Injection"
] | [
"Definition:Injection",
"Definition:Linearly Independent",
"Category:Module Theory"
] |
proofwiki-12346 | Free Module is Isomorphic to Free Module on Set | Let $M$ be a unitary $R$-module.
Let $\BB = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by Universal Property of Free Module on Set.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\BB$ is a basis of $M$}}
{{item|(2):|$\Psi$ is an isomorphism}}
{{end-ite... | Follows directly from:
:Characterisation of Linearly Independent Set through Free Module Indexed by Set
:Characterisation of Spanning Set through Free Module Indexed by Set.
{{qed}}
Category:Free Modules
Category:Module Theory
ajf3mybiohgd4pj9cxrm4lzggyg36t6 | Let $M$ be a [[Definition:Unitary Module|unitary $R$-module]].
Let $\BB = \family {b_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by [[Universal Property of Free Module on Set]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\BB$ ... | Follows directly from:
:[[Characterisation of Linearly Independent Set through Free Module Indexed by Set]]
:[[Characterisation of Spanning Set through Free Module Indexed by Set]].
{{qed}}
[[Category:Free Modules]]
[[Category:Module Theory]]
ajf3mybiohgd4pj9cxrm4lzggyg36t6 | Free Module is Isomorphic to Free Module on Set | https://proofwiki.org/wiki/Free_Module_is_Isomorphic_to_Free_Module_on_Set | https://proofwiki.org/wiki/Free_Module_is_Isomorphic_to_Free_Module_on_Set | [
"Free Modules",
"Module Theory"
] | [
"Definition:Unitary Module over Ring",
"Definition:Indexing Set/Family",
"Universal Property of Free Module on Set",
"Definition:Basis of Module",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism"
] | [
"Characterisation of Linearly Independent Set through Free Module Indexed by Set",
"Characterisation of Spanning Set through Free Module Indexed by Set",
"Category:Free Modules",
"Category:Module Theory"
] |
proofwiki-12347 | Direct Product of Modules is Module | Let $R$ be a ring.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a module. | From External Direct Product of Abelian Groups is Abelian Group it follows that $(M,+)$ is an abelian group.
We need to show that:
$\forall x, y, \in M, \forall \lambda, \mu \in R$:
: $(1): \quad \lambda \circ \paren {x + y} = \paren {\lambda \circ x} + \paren {\lambda \circ y}$
: $(2): \quad \paren {\lambda +_R \mu} \... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Module over Ring|$R$-modules]].
Let $\struct {M, +, \circ}$ be their [[Definition:Module Direct Product|direct product]].
Then $\struct {... | From [[External Direct Product of Abelian Groups is Abelian Group]] it follows that $(M,+)$ is an [[Definition:Abelian Group|abelian group]].
We need to show that:
$\forall x, y, \in M, \forall \lambda, \mu \in R$:
: $(1): \quad \lambda \circ \paren {x + y} = \paren {\lambda \circ x} + \paren {\lambda \circ y}$
: $... | Direct Product of Modules is Module | https://proofwiki.org/wiki/Direct_Product_of_Modules_is_Module | https://proofwiki.org/wiki/Direct_Product_of_Modules_is_Module | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Indexing Set/Family",
"Definition:Module over Ring",
"Definition:Module Direct Product",
"Definition:Module over Ring"
] | [
"External Direct Product of Abelian Groups is Abelian Group",
"Definition:Abelian Group"
] |
proofwiki-12348 | Direct Product of Modules is Module | Let $R$ be a ring.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a module. | This is a special case of Direct Product of Modules is Module. | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Module over Ring|$R$-modules]].
Let $\struct {M, +, \circ}$ be their [[Definition:Module Direct Product|direct product]].
Then $\struct {... | This is a special case of [[Direct Product of Modules is Module]]. | Finite Direct Product of Modules is Module/Proof 1 | https://proofwiki.org/wiki/Direct_Product_of_Modules_is_Module | https://proofwiki.org/wiki/Finite_Direct_Product_of_Modules_is_Module/Proof_1 | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Indexing Set/Family",
"Definition:Module over Ring",
"Definition:Module Direct Product",
"Definition:Module over Ring"
] | [
"Direct Product of Modules is Module"
] |
proofwiki-12349 | Direct Product of Modules is Module | Let $R$ be a ring.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a module. | === {{Module-axiom|1|nolink}} ===
Let $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in G$.
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren {x + y}
| r = \lambda \circ \paren {\tuple {x_1, x_2, \ldots, x_n} + \tuple {y_1, y_2, \ldots, y_n} }
| c =
}}
{{eqn | r = \lambda \circ \tupl... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Module over Ring|$R$-modules]].
Let $\struct {M, +, \circ}$ be their [[Definition:Module Direct Product|direct product]].
Then $\struct {... | === {{Module-axiom|1|nolink}} ===
Let $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in G$.
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren {x + y}
| r = \lambda \circ \paren {\tuple {x_1, x_2, \ldots, x_n} + \tuple {y_1, y_2, \ldots, y_n} }
| c =
}}
{{eqn | r = \lambda \circ \tu... | Finite Direct Product of Modules is Module/Proof 2 | https://proofwiki.org/wiki/Direct_Product_of_Modules_is_Module | https://proofwiki.org/wiki/Finite_Direct_Product_of_Modules_is_Module/Proof_2 | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Indexing Set/Family",
"Definition:Module over Ring",
"Definition:Module Direct Product",
"Definition:Module over Ring"
] | [] |
proofwiki-12350 | Direct Product of Unitary Modules is Unitary Module | Let $R$ be a ring with unity whose unity is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of unitary $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a unitary $R$-module. | From Direct Product of Modules is Module, $M$ is an $R$-module.
It remains to verify that:
:$\forall x \in M: 1_R \circ x = x$
By the definition of direct product, $\circ$ is a ring action induced on $M$ by $\family {\circ_i}_{i \mathop \in I}$:
:$\forall r \in R: r \circ \family {m_i}_{i \mathop \in I} = \family {r \c... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Unitary Module over Ring|unitary $R$-modules]].
Let $\struct {M, +, \circ}$ be their [... | From [[Direct Product of Modules is Module]], $M$ is an [[Definition:Module over Ring|$R$-module]].
It remains to verify that:
:$\forall x \in M: 1_R \circ x = x$
By the definition of [[Definition:Module Direct Product|direct product]], $\circ$ is a [[Definition:Left Linear Ring Action|ring action]] [[Definition:Op... | Direct Product of Unitary Modules is Unitary Module | https://proofwiki.org/wiki/Direct_Product_of_Unitary_Modules_is_Unitary_Module | https://proofwiki.org/wiki/Direct_Product_of_Unitary_Modules_is_Unitary_Module | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Indexing Set/Family",
"Definition:Unitary Module over Ring",
"Definition:Module Direct Product",
"Definition:Unitary Module over Ring"
] | [
"Direct Product of Modules is Module",
"Definition:Module over Ring",
"Definition:Module Direct Product",
"Definition:Linear Ring Action/Left",
"Definition:Operation Induced by Direct Product",
"Category:Module Theory",
"Category:Direct Products"
] |
proofwiki-12351 | Direct Product of Unitary Modules is Unitary Module | Let $R$ be a ring with unity whose unity is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of unitary $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a unitary $R$-module. | This is a special case of Direct Product of Unitary Modules is Unitary Module. | Let $R$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Unitary Module over Ring|unitary $R$-modules]].
Let $\struct {M, +, \circ}$ be their [... | This is a special case of [[Direct Product of Unitary Modules is Unitary Module]]. | Finite Direct Product of Unitary Modules is Unitary Module/Proof 1 | https://proofwiki.org/wiki/Direct_Product_of_Unitary_Modules_is_Unitary_Module | https://proofwiki.org/wiki/Finite_Direct_Product_of_Unitary_Modules_is_Unitary_Module/Proof_1 | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Indexing Set/Family",
"Definition:Unitary Module over Ring",
"Definition:Module Direct Product",
"Definition:Unitary Module over Ring"
] | [
"Direct Product of Unitary Modules is Unitary Module"
] |
proofwiki-12352 | Direct Product of Unitary Modules is Unitary Module | Let $R$ be a ring with unity whose unity is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of unitary $R$-modules.
Let $\struct {M, +, \circ}$ be their direct product.
Then $\struct {M, +, \circ}$ is a unitary $R$-module. | From Finite Direct Product of Modules is Module we have that $G$ is a module.
It remains to be shown that:
:$\forall x \in G: 1_R \circ x = x$
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.
Then:
{{begin-eqn}}
{{eqn | l = 1_R \circ x
| r = 1_R \circ \tuple {x_1, x_2, \ldots, x_n}
| c =
}}
{{eqn | r = \tup... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Unitary Module over Ring|unitary $R$-modules]].
Let $\struct {M, +, \circ}$ be their [... | From [[Finite Direct Product of Modules is Module]] we have that $G$ is a [[Definition:Module over Ring|module]].
It remains to be shown that:
:$\forall x \in G: 1_R \circ x = x$
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.
Then:
{{begin-eqn}}
{{eqn | l = 1_R \circ x
| r = 1_R \circ \tuple {x_1, x_2, \ld... | Finite Direct Product of Unitary Modules is Unitary Module/Proof 2 | https://proofwiki.org/wiki/Direct_Product_of_Unitary_Modules_is_Unitary_Module | https://proofwiki.org/wiki/Finite_Direct_Product_of_Unitary_Modules_is_Unitary_Module/Proof_2 | [
"Module Theory",
"Direct Products"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Indexing Set/Family",
"Definition:Unitary Module over Ring",
"Definition:Module Direct Product",
"Definition:Unitary Module over Ring"
] | [
"Finite Direct Product of Modules is Module",
"Definition:Module over Ring"
] |
proofwiki-12353 | Auxiliary Relation Image of Element is Upper Section | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $R$ be an auxiliary relation on $S$.
Let $x \in S$.
Then $\map R x$ is an upper section
where $\map R x$ denotes the image of $x$ under $R$. | Let $a \in \map R x, b \in S$ such that
:$a \preceq b$
By definition of $R$-image of element:
:$\tuple {x, a} \in R$
By definition of reflexivity:
:$x \preceq x$
By definition of auxiliary relation:
:$\tuple {x, b} \in R$
Thus by definition of $R$-image of element:
:$b \in \map R x$
{{qed}} | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $R$ be an [[Definition:Auxiliary Relation|auxiliary relation]] on $S$.
Let $x \in S$.
Then $\map R x$ is an [[Definition:Upper Section|upper section]]
where $\map R x$ denotes the [[Definition:Image of Element under Relation|image]] o... | Let $a \in \map R x, b \in S$ such that
:$a \preceq b$
By definition of [[Definition:Image of Element under Relation|$R$-image of element]]:
:$\tuple {x, a} \in R$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \preceq x$
By definition of [[Definition:Auxiliary Relation|auxiliary relation]]:
:$\tuple {... | Auxiliary Relation Image of Element is Upper Section | https://proofwiki.org/wiki/Auxiliary_Relation_Image_of_Element_is_Upper_Section | https://proofwiki.org/wiki/Auxiliary_Relation_Image_of_Element_is_Upper_Section | [
"Auxiliary Relations"
] | [
"Definition:Ordered Set",
"Definition:Auxiliary Relation",
"Definition:Upper Section",
"Definition:Image (Set Theory)/Relation/Element"
] | [
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Reflexivity",
"Definition:Auxiliary Relation",
"Definition:Image (Set Theory)/Relation/Element"
] |
proofwiki-12354 | Multiplicative Auxiliary Relation iff Congruent | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice.
Let $\RR$ be an auxiliary relation on $S$.
Then $\RR$ is multiplicative {{iff}}:
:$\forall a, b, x, y \in S: \tuple {a, x}, \tuple {b, y} \in \RR \implies \tuple {a \wedge b, x \wedge y} \in \RR$
That is {{iff}} $\RR$ is a congruence relation for ... | === Sufficient Condition ===
Let $\RR$ be multiplicative.
Let $a, b, x, y \in S$ such that
:$\tuple {a, x}, \tuple {b, y} \in \RR$
By Meet Precedes Operands:
:$a \wedge b \preceq a$ and $a \wedge b \preceq b$
By definition of reflexivity:
:$x \preceq x$ and $y \preceq y$
By definition of auxiliary relation:
:$\tuple {a... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Lattice (Order Theory)|lattice]].
Let $\RR$ be an [[Definition:Auxiliary Relation|auxiliary relation]] on $S$.
Then $\RR$ is [[Definition:Multiplicative Relation|multiplicative]] {{iff}}:
:$\forall a, b, x,... | === Sufficient Condition ===
Let $\RR$ be [[Definition:Multiplicative Relation|multiplicative]].
Let $a, b, x, y \in S$ such that
:$\tuple {a, x}, \tuple {b, y} \in \RR$
By [[Meet Precedes Operands]]:
:$a \wedge b \preceq a$ and $a \wedge b \preceq b$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \pr... | Multiplicative Auxiliary Relation iff Congruent | https://proofwiki.org/wiki/Multiplicative_Auxiliary_Relation_iff_Congruent | https://proofwiki.org/wiki/Multiplicative_Auxiliary_Relation_iff_Congruent | [
"Order Theory",
"Auxiliary Relations"
] | [
"Definition:Bounded Below Set",
"Definition:Lattice (Order Theory)",
"Definition:Auxiliary Relation",
"Definition:Multiplicative Relation",
"Definition:Congruence Relation"
] | [
"Definition:Multiplicative Relation",
"Meet Precedes Operands",
"Definition:Reflexivity",
"Definition:Auxiliary Relation",
"Definition:Multiplicative Relation"
] |
proofwiki-12355 | Cube which is One Less than a Square | $8$ is the only cube number which is $1$ less than a square:
:$2^3 + 1 = 3^2$ | This is a specific instance of 1 plus Perfect Power is not Prime Power except for 9.
{{qed}} | $8$ is the only [[Definition:Cube Number|cube number]] which is $1$ less than a [[Definition:Square Number|square]]:
:$2^3 + 1 = 3^2$ | This is a specific instance of [[1 plus Perfect Power is not Prime Power except for 9]].
{{qed}} | Cube which is One Less than a Square | https://proofwiki.org/wiki/Cube_which_is_One_Less_than_a_Square | https://proofwiki.org/wiki/Cube_which_is_One_Less_than_a_Square | [
"8",
"9",
"Square Numbers",
"Cube Numbers"
] | [
"Definition:Cube Number",
"Definition:Square Number"
] | [
"1 plus Perfect Power is not Prime Power except for 9"
] |
proofwiki-12356 | Prime Powers Differing by One | $8$ and $9$ are the only powers of prime numbers which differ by exactly $1$:
:$2^3 + 1 = 3^2$ | This is a direct consequence of 1 plus Perfect Power is not Prime Power except for 9.
{{qed}} | $8$ and $9$ are the only [[Definition:Integer Power|powers]] of [[Definition:Prime Number|prime numbers]] which differ by exactly $1$:
:$2^3 + 1 = 3^2$ | This is a direct consequence of [[1 plus Perfect Power is not Prime Power except for 9]].
{{qed}} | Prime Powers Differing by One | https://proofwiki.org/wiki/Prime_Powers_Differing_by_One | https://proofwiki.org/wiki/Prime_Powers_Differing_by_One | [
"8",
"9",
"Powers"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Prime Number"
] | [
"1 plus Perfect Power is not Prime Power except for 9"
] |
proofwiki-12357 | Monoid Ring of Commutative Monoid over Commutative Ring is Commutative | Let $R$ be a commutative ring.
Let $G$ be a commutative monoid.
Let $\sqbrk R G$ be the monoid ring of $G$ over $R$.
Then $\sqbrk R G$ is commutative. | {{proof wanted}}
Category:Commutative Monoids
Category:Commutative Rings
d4keyldaa78pzil8a5k8ky59lhnxm9a | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $G$ be a [[Definition:Commutative Monoid|commutative monoid]].
Let $\sqbrk R G$ be the [[Definition:Monoid Ring|monoid ring]] of $G$ over $R$.
Then $\sqbrk R G$ is [[Definition:Commutative Ring|commutative]]. | {{proof wanted}}
[[Category:Commutative Monoids]]
[[Category:Commutative Rings]]
d4keyldaa78pzil8a5k8ky59lhnxm9a | Monoid Ring of Commutative Monoid over Commutative Ring is Commutative | https://proofwiki.org/wiki/Monoid_Ring_of_Commutative_Monoid_over_Commutative_Ring_is_Commutative | https://proofwiki.org/wiki/Monoid_Ring_of_Commutative_Monoid_over_Commutative_Ring_is_Commutative | [
"Commutative Monoids",
"Commutative Rings"
] | [
"Definition:Commutative Ring",
"Definition:Commutative Monoid",
"Definition:Monoid Ring",
"Definition:Commutative Ring"
] | [
"Category:Commutative Monoids",
"Category:Commutative Rings"
] |
proofwiki-12358 | 3 Non-Parallel Planes divide Space into 8 | Let $3$ planes which are pairwise non-parallel be constructed in ordinary $3$-dimensional space.
Then that space is divided into $8$ parts by those planes. | {{ProofWanted|Intuitively obvious but needs a run-up<br/>It's not actually even true -- consider the case where all $3$ lines of intersection of the $3$ planes are parallel. Needs to be reworded, presumably just means the lines of intersection are pairwise non-parallel as well.}} | Let $3$ [[Definition:Plane|planes]] which are pairwise non-[[Definition:Parallel Planes|parallel]] be constructed in [[Definition:Ordinary Space|ordinary]] [[Definition:Dimension (Geometry)|$3$-dimensional space]].
Then that [[Definition:Ordinary Space|space]] is divided into $8$ parts by those [[Definition:Plane|pla... | {{ProofWanted|Intuitively obvious but needs a run-up<br/>It's not actually even true -- consider the case where all $3$ lines of intersection of the $3$ planes are parallel. Needs to be reworded, presumably just means the lines of intersection are pairwise non-parallel as well.}} | 3 Non-Parallel Planes divide Space into 8 | https://proofwiki.org/wiki/3_Non-Parallel_Planes_divide_Space_into_8 | https://proofwiki.org/wiki/3_Non-Parallel_Planes_divide_Space_into_8 | [
"8",
"Solid Geometry",
"Planes"
] | [
"Definition:Plane Surface",
"Definition:Parallel (Geometry)/Planes",
"Definition:Ordinary Space",
"Definition:Dimension (Geometry)",
"Definition:Ordinary Space",
"Definition:Plane Surface"
] | [] |
proofwiki-12359 | Divisibility by 8 | An integer $N$ expressed in decimal notation is divisible by $8$ {{iff}} the $3$ {{LSD}}s of $N$ form a $3$-digit integer divisible by $8$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $8$
{{iff}}:
:$100 a_2 + 10 a_1 + a_0$ is divisible by $8$. | Let $N$ be divisible by $8$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 8
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 8
}}
{{eqn | ll= \leadstoandfrom
| l = 100 a_2 + 10 a_1 + a_0 + 10^3 \sum_{k \mathop = 3}^n a_k ... | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $8$ {{iff}} the $3$ {{LSD}}s of $N$ form a $3$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] [[Definition:Divisor of Integer|divisible]] by $8$.
That is:
:$... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $8$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 8
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 8
}}
{{eqn | ll= \leadstoandfrom
| l = 100 a_2 + 10 a_1 + a_0... | Divisibility by 8 | https://proofwiki.org/wiki/Divisibility_by_8 | https://proofwiki.org/wiki/Divisibility_by_8 | [
"Divisibility Tests",
"8"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12360 | Closed Form for Octagonal Numbers | The closed-form expression for the $n$th octagonal number is:
:$O_n = n \paren {3 n - 2}$ | Octagonal numbers are $k$-gonal numbers where $k = 8$.
From Closed Form for Polygonal Numbers we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = O_n
| r = \frac n 2 \paren {\paren {8 - 2} n - 8 + 4}
| c = Closed Form for Polygonal Numbers
}}
{{eqn ... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Octagonal Number|octagonal number]] is:
:$O_n = n \paren {3 n - 2}$ | [[Definition:Octagonal Number|Octagonal numbers]] are [[Definition:Polygonal Number|$k$-gonal numbers]] where $k = 8$.
From [[Closed Form for Polygonal Numbers]] we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = O_n
| r = \frac n 2 \paren {\paren {8 ... | Closed Form for Octagonal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Octagonal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Octagonal_Numbers | [
"Octagonal Numbers",
"Closed Forms"
] | [
"Definition:Closed Form Expression",
"Definition:Octagonal Number"
] | [
"Definition:Octagonal Number",
"Definition:Polygonal Number",
"Closed Form for Polygonal Numbers",
"Closed Form for Polygonal Numbers"
] |
proofwiki-12361 | No Perfect Magic Cube of Order Less than 5 Exists | Apart from the trivial order $1$ case, no perfect magic cube exists whose order is $4$ or less. | === Order $2$ ===
Consider a layer of the order $2$ perfect magic cube:
:<nowiki>$\begin{array}{|c|c|}
\hline a & b \\
\hline c & d \\
\hline
\end{array}$</nowiki>
Then we must have $a + b = a + c$.
So $b = c$, so they are not distinct, so this array cannot be a layer of a perfect magic cube.
{{qed|lemma}} | Apart from the trivial [[Definition:Order of Magic Cube|order $1$]] case, no [[Definition:Perfect Magic Cube|perfect magic cube]] exists whose [[Definition:Order of Magic Cube|order]] is $4$ or less. | === Order $2$ ===
Consider a layer of the [[Definition:Order of Magic Cube|order $2$]] [[Definition:Perfect Magic Cube|perfect magic cube]]:
:<nowiki>$\begin{array}{|c|c|}
\hline a & b \\
\hline c & d \\
\hline
\end{array}$</nowiki>
Then we must have $a + b = a + c$.
So $b = c$, so they are not distinct, so this ... | No Perfect Magic Cube of Order Less than 5 Exists | https://proofwiki.org/wiki/No_Perfect_Magic_Cube_of_Order_Less_than_5_Exists | https://proofwiki.org/wiki/No_Perfect_Magic_Cube_of_Order_Less_than_5_Exists | [
"Perfect Magic Cubes"
] | [
"Definition:Magic Cube/Order",
"Definition:Perfect Magic Cube",
"Definition:Magic Cube/Order"
] | [
"Definition:Magic Cube/Order",
"Definition:Perfect Magic Cube",
"Definition:Perfect Magic Cube",
"Definition:Magic Cube/Order",
"Definition:Perfect Magic Cube",
"Definition:Perfect Magic Cube",
"Definition:Perfect Magic Cube",
"Definition:Magic Cube/Order",
"Definition:Perfect Magic Cube",
"Defini... |
proofwiki-12362 | Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part | Let $z \in \C$ be a complex number.
Let $t > 0$ be wholly real.
Let $t^z$ be $t$ to the power of $z$ defined on its principal branch.
Then:
:$\cmod {t^z} = t^{\map \Re z}$ | {{begin-eqn}}
{{eqn | l = \cmod {t^z}
| r = \cmod {t^{\map \Re z + i \map \Im z} }
}}
{{eqn | r = \cmod {t^{\map \Re z} t^{i \map \Im z} }
| c = Sum of Complex Indices of Real Number
}}
{{eqn | r = \cmod {t^{\map \Re z} } \cmod {t^{i \map \Im z} }
| c = Complex Modulus of Product of Complex Numbers
}}... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $t > 0$ be [[Definition:Wholly Real|wholly real]].
Let $t^z$ be [[Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base|$t$ to the power of $z$ defined on its principal branch]].
Then:
:$\cmod {t^z} = t^{\map \Re z}$ | {{begin-eqn}}
{{eqn | l = \cmod {t^z}
| r = \cmod {t^{\map \Re z + i \map \Im z} }
}}
{{eqn | r = \cmod {t^{\map \Re z} t^{i \map \Im z} }
| c = [[Sum of Complex Indices of Real Number]]
}}
{{eqn | r = \cmod {t^{\map \Re z} } \cmod {t^{i \map \Im z} }
| c = [[Complex Modulus of Product of Complex Numb... | Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part | https://proofwiki.org/wiki/Modulus_of_Positive_Real_Number_to_Complex_Power_is_Positive_Real_Number_to_Power_of_Real_Part | https://proofwiki.org/wiki/Modulus_of_Positive_Real_Number_to_Complex_Power_is_Positive_Real_Number_to_Power_of_Real_Part | [
"Complex Modulus",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Wholly Real",
"Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base"
] | [
"Sum of Complex Indices of Real Number",
"Complex Modulus of Product of Complex Numbers",
"Modulus of Exponential of Imaginary Number is One",
"Power of Positive Real Number is Positive/Real Number",
"Category:Complex Modulus",
"Category:Complex Analysis"
] |
proofwiki-12363 | Characterization of Pseudoprime Element when Way Below Relation is Multiplicative | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below continuous lattice such that
:$\ll$ is multiplicative relation
where $\ll$ denotes the way below relation of $L$.
Let $p \in S$.
Then $p$ is pseudoprime element {{iff}}
:$\forall a, b \in S: a \wedge b \ll p \implies a \preceq p \lor b \preceq p$ | === Sufficient Condition ===
Let $p$ be pseudoprime element.
Let $a, b \in S$ such that
:$a \wedge b \ll p$
By definition of meet:
:$\inf \left\{ {a, b}\right\} \ll p$
By Characterization of Pseudoprime Element by Finite Infima:
:$\exists c \in \left\{ {a, b}\right\}: c \preceq p$
Thus
:$a \preceq p$ or $b \preceq p$
{... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Lattice (Order Theory)|lattice]] such that
:$\ll$ is [[Definition:Multiplicative Relation|multiplicative relation]]
where $\ll$ denotes the [[Definition:Element ... | === Sufficient Condition ===
Let $p$ be [[Definition:Pseudoprime (Order Theory)|pseudoprime element]].
Let $a, b \in S$ such that
:$a \wedge b \ll p$
By definition of [[Definition:Meet (Order Theory)|meet]]:
:$\inf \left\{ {a, b}\right\} \ll p$
By [[Characterization of Pseudoprime Element by Finite Infima]]:
:$\exi... | Characterization of Pseudoprime Element when Way Below Relation is Multiplicative | https://proofwiki.org/wiki/Characterization_of_Pseudoprime_Element_when_Way_Below_Relation_is_Multiplicative | https://proofwiki.org/wiki/Characterization_of_Pseudoprime_Element_when_Way_Below_Relation_is_Multiplicative | [
"Prime Elements",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Continuous Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Multiplicative Relation",
"Definition:Element is Way Below",
"Definition:Pseudoprime (Order Theory)"
] | [
"Definition:Pseudoprime (Order Theory)",
"Definition:Meet (Order Theory)",
"Characterization of Pseudoprime Element by Finite Infima",
"Definition:Pseudoprime (Order Theory)"
] |
proofwiki-12364 | Number of Distinct Deltahedra is Unlimited | There are an unlimited number of distinct deltahedra. | To any deltahedron, it is possible to attach a regular tetrahedron to any of its faces to create a new deltahedron with $2$ more faces. | There are an unlimited number of [[Definition:Distinct|distinct]] [[Definition:Deltahedron|deltahedra]]. | To any [[Definition:Deltahedron|deltahedron]], it is possible to attach a [[Definition:Regular Tetrahedron|regular tetrahedron]] to any of its [[Definition:Face of Polyhedron|faces]] to create a new [[Definition:Deltahedron|deltahedron]] with $2$ more [[Definition:Face of Polyhedron|faces]]. | Number of Distinct Deltahedra is Unlimited | https://proofwiki.org/wiki/Number_of_Distinct_Deltahedra_is_Unlimited | https://proofwiki.org/wiki/Number_of_Distinct_Deltahedra_is_Unlimited | [
"Deltahedra"
] | [
"Definition:Distinct",
"Definition:Deltahedron"
] | [
"Definition:Deltahedron",
"Definition:Tetrahedron/Regular",
"Definition:Polyhedron/Face",
"Definition:Deltahedron",
"Definition:Polyhedron/Face"
] |
proofwiki-12365 | Way Below Relation is Multiplicative implies Pseudoprime Element is Prime | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below continuous lattice such that
:$\ll$ is multiplicative relation
where $\ll$ denotes the way below relation of $L$.
Let $p \in S$.
Then $p$ is a pseudoprime element is a prime element. | Let $p$ be a pseudoprime element.
{{AimForCont}}:
:$p$ is not a prime element.
By definition of prime element:
:$\exists x, y \in S: x \wedge y \preceq p$ and $x \npreceq p$ and $y \npreceq p$
By definition of continuous:
:$\forall z \in S: z^\ll$ is directed.
and
:$L$ satisfies the axiom of approximation.
By Axiom of ... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Lattice (Order Theory)|lattice]] such that
:$\ll$ is [[Definition:Multiplicative Relation|multiplicative relation]]
where $\ll$ denotes the [[Definition:Element ... | Let $p$ be a [[Definition:Pseudoprime (Order Theory)|pseudoprime element]].
{{AimForCont}}:
:$p$ is not a [[Definition:Prime Element (Order Theory)|prime element]].
By definition of [[Definition:Prime Element (Order Theory)|prime element]]:
:$\exists x, y \in S: x \wedge y \preceq p$ and $x \npreceq p$ and $y \nprece... | Way Below Relation is Multiplicative implies Pseudoprime Element is Prime | https://proofwiki.org/wiki/Way_Below_Relation_is_Multiplicative_implies_Pseudoprime_Element_is_Prime | https://proofwiki.org/wiki/Way_Below_Relation_is_Multiplicative_implies_Pseudoprime_Element_is_Prime | [
"Prime Elements",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Continuous Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Multiplicative Relation",
"Definition:Element is Way Below",
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Element (Order Theory)"
] | [
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Continuous Ordered Set",
"Definition:Directed Subset",
"Axiom:Axiom of Approximation",
"Axiom of Approximation in Up-Complete Semilattice",
"Way Below Relation is ... |
proofwiki-12366 | If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below continuous distributive lattice.
Let every element $p \in S$: $p$ is pseudoprime $\implies p$ is prime.
Then $\ll$ is multiplicative
where $\ll$ denotes the way below relation of $L$. | Let $a, x, y \in S$ such that:
:$a \ll x$ and $a \ll y$
{{AimForCont}}:
:$a \not\ll x \wedge y$
We will prove that:
:$\forall z \in S: z \in \paren {x \wedge y}^\ll \implies z \notin a^\succeq$
Let $z \in S$ such that:
:$z \in \paren {x \wedge y}^\ll$
By definition of way below closure:
:$z \ll x \wedge y$
{{AimForCont... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Distributive Lattice|distributive lattice]].
Let every [[Definition:Element|element]] $p \in S$: $p$ is [[Definition:Pseudoprime (Order Theory)|pseudoprime]] $\... | Let $a, x, y \in S$ such that:
:$a \ll x$ and $a \ll y$
{{AimForCont}}:
:$a \not\ll x \wedge y$
We will prove that:
:$\forall z \in S: z \in \paren {x \wedge y}^\ll \implies z \notin a^\succeq$
Let $z \in S$ such that:
:$z \in \paren {x \wedge y}^\ll$
By definition of [[Definition:Way Below Closure|way below closur... | If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative | https://proofwiki.org/wiki/If_Every_Element_Pseudoprime_is_Prime_then_Way_Below_Relation_is_Multiplicative | https://proofwiki.org/wiki/If_Every_Element_Pseudoprime_is_Prime_then_Way_Below_Relation_is_Multiplicative | [
"Prime Elements",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Continuous Ordered Set",
"Definition:Distributive Lattice",
"Definition:Element",
"Definition:Pseudoprime (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Multiplicative Relation",
"Definition:Element is Way Below"
] | [
"Definition:Way Below Closure",
"Definition:Upper Closure/Element",
"Preceding and Way Below implies Way Below",
"Definition:Contradiction",
"Definition:Empty Set",
"Definition:Set Intersection",
"Way Below Closure is Ideal in Bounded Below Join Semilattice",
"Definition:Ideal (Order Theory)",
"Uppe... |
proofwiki-12367 | Upper Closure of Element is Way Below Open Filter iff Element is Compact | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $x \in S$.
Then:
:$x^\succeq$ is a way below open filter on $L$
{{iff}}
:$x$ is compact | === Sufficient Condition ===
Suppose:
:$x^\succeq$ is a way below open filter on $L$
By definitions of upper closure of element and reflexivity:
:$x \in x^\succeq$
By definition of way below open:
:$\exists y \in x^\succeq: y \ll x$
By definition of upper closure of element:
:$x \preceq y$
By Preceding and Way Below im... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $x \in S$.
Then:
:$x^\succeq$ is a [[Definition:Way Below Open|way below open]] [[Definition:Filter|filter]] on $L$
{{iff}}
:$x$ is [[Definition:Compact Element|compact]] | === Sufficient Condition ===
Suppose:
:$x^\succeq$ is a [[Definition:Way Below Open|way below open]] [[Definition:Filter|filter]] on $L$
By definitions of [[Definition:Upper Closure of Element|upper closure of element]] and [[Definition:Reflexivity|reflexivity]]:
:$x \in x^\succeq$
By definition of [[Definition:Way ... | Upper Closure of Element is Way Below Open Filter iff Element is Compact | https://proofwiki.org/wiki/Upper_Closure_of_Element_is_Way_Below_Open_Filter_iff_Element_is_Compact | https://proofwiki.org/wiki/Upper_Closure_of_Element_is_Way_Below_Open_Filter_iff_Element_is_Compact | [
"Way Below Relation"
] | [
"Definition:Complete Lattice",
"Definition:Way Below Open",
"Definition:Filter",
"Definition:Compact Element"
] | [
"Definition:Way Below Open",
"Definition:Filter",
"Definition:Upper Closure/Element",
"Definition:Reflexivity",
"Definition:Way Below Open",
"Definition:Upper Closure/Element",
"Preceding and Way Below implies Way Below",
"Definition:Reflexivity",
"Definition:Compact Element",
"Definition:Compact ... |
proofwiki-12368 | 9 is Only Square which is Sum of 2 Consecutive Positive Cubes | Discounting the trivial solution:
:$1^2 = 1 = 0^3 + 1^3$
$9$ is the only square number which is the sum of $2$ consecutive positive cube numbers:
:$3^2 = 9 = 1^3 + 2^3$ | The expression for the $n$th square number is:
:$n^2$
for $n \in \Z$.
The expression for the $n$th cube number is:
:$n^3$
again, for $n \in \Z$.
Therefore the closed-form expression for the $n$th sum of two consecutive cubes is:
:$n^3 + \paren {n + 1}^3$
This simplifies by {{Corollary|Cube of Sum}} to:
:$2 n^3 + 3 n^2 ... | Discounting the trivial solution:
:$1^2 = 1 = 0^3 + 1^3$
$9$ is the only [[Definition:Square Number|square number]] which is the sum of $2$ consecutive [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cube numbers]]:
:$3^2 = 9 = 1^3 + 2^3$ | The expression for the $n$th [[Definition:Square Number|square number]] is:
:$n^2$
for $n \in \Z$.
The expression for the $n$th [[Definition:Cube Number|cube number]] is:
:$n^3$
again, for $n \in \Z$.
Therefore the [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th sum of two consecutive [[D... | 9 is Only Square which is Sum of 2 Consecutive Positive Cubes | https://proofwiki.org/wiki/9_is_Only_Square_which_is_Sum_of_2_Consecutive_Positive_Cubes | https://proofwiki.org/wiki/9_is_Only_Square_which_is_Sum_of_2_Consecutive_Positive_Cubes | [
"9",
"Square Numbers",
"Cube Numbers",
"Sums of Cubes"
] | [
"Definition:Square Number",
"Definition:Positive/Integer",
"Definition:Cube Number"
] | [
"Definition:Square Number",
"Definition:Cube Number",
"Definition:Closed Form Expression",
"Definition:Cube Number",
"Integral Points of Elliptic Curve y^2 = x^3+3x",
"Definition:Integer"
] |
proofwiki-12369 | If Compact Between then Way Below | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $x, k, y \in S$ such that:
:$x \preceq k$ and $k \preceq y$ and $k \in \map K L$
where $\map K L$ denotes the compact subset of $L$.
Then $x \ll y$
where $\ll$ denotes the way below relation. | By definition of compact subset:
:$k$ is compact.
By definition of compact:
:$k \ll k$
Thus by Preceding and Way Below implies Way Below:
:$x \ll y$
{{qed}} | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $x, k, y \in S$ such that:
:$x \preceq k$ and $k \preceq y$ and $k \in \map K L$
where $\map K L$ denotes the [[Definition:Compact Subset of Lattice|compact subset]] of $L$.
Then $x \ll y$
where $\ll$ denotes the... | By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$k$ is [[Definition:Compact Element|compact]].
By definition of [[Definition:Compact Element|compact]]:
:$k \ll k$
Thus by [[Preceding and Way Below implies Way Below]]:
:$x \ll y$
{{qed}} | If Compact Between then Way Below | https://proofwiki.org/wiki/If_Compact_Between_then_Way_Below | https://proofwiki.org/wiki/If_Compact_Between_then_Way_Below | [
"Way Below Relation"
] | [
"Definition:Complete Lattice",
"Definition:Compact Subset of Lattice",
"Definition:Element is Way Below"
] | [
"Definition:Compact Subset of Lattice",
"Definition:Compact Element",
"Definition:Compact Element",
"Preceding and Way Below implies Way Below"
] |
proofwiki-12370 | Nine Regular Polyhedra | There exist $9$ regular polyhedra. | From Five Platonic Solids, there exist $5$ regular polyhedra which are convex:
:the regular tetrahedron
:the cube
:the regular octahedron
:the regular dodecahedron
:the regular icosahedron.
From Four Kepler-Poinsot Polyhedra:
{{:Four Kepler-Poinsot Polyhedra}}
All $4$ of the above are regular polyhedra which are non-co... | There exist $9$ [[Definition:Regular Polyhedron|regular polyhedra]]. | From [[Five Platonic Solids]], there exist $5$ [[Definition:Regular Polyhedron|regular polyhedra]] which are [[Definition:Convex Polyhedron|convex]]:
:the [[Definition:Regular Tetrahedron|regular tetrahedron]]
:the [[Definition:Cube (Geometry)|cube]]
:the [[Definition:Regular Octahedron|regular octahedron]]
:the [[Defi... | Nine Regular Polyhedra | https://proofwiki.org/wiki/Nine_Regular_Polyhedra | https://proofwiki.org/wiki/Nine_Regular_Polyhedra | [
"Regular Polyhedra"
] | [
"Definition:Regular Polyhedron"
] | [
"Five Platonic Solids",
"Definition:Regular Polyhedron",
"Definition:Convex Polyhedron",
"Definition:Tetrahedron/Regular",
"Definition:Cube/Geometry",
"Definition:Octahedron/Regular",
"Definition:Dodecahedron/Regular",
"Definition:Icosahedron/Regular",
"Four Kepler-Poinsot Polyhedra",
"Definition:... |
proofwiki-12371 | Way Below is Congruent for Join | Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.
Then $\ll$ is congruence relation for $\vee$:
:$\forall a, b, x, y \in S: a \ll x \land b \ll y \implies a \vee b \ll x \vee y$
where $\ll$ denotes the way below relation. | Let $a, b, x, y \in S$ such that
$a \ll x$ and $b \ll y$
By Join Succeeds Operands:
:$x \preceq x \vee y$ and $y \preceq x \vee y$
By Preceding and Way Below implies Way Below and definition of reflexivity:
:$a \ll x \vee y$ and $b \ll x \vee y$
Thus by Join is Way Below if Operands are Way Below:
:$a \vee b \ll x \vee... | Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Join Semilattice|join semilattice]].
Then $\ll$ is [[Definition:Congruence Relation|congruence relation]] for $\vee$:
:$\forall a, b, x, y \in S: a \ll x \land b \ll y \implies a \vee b \ll x \vee y$
where $\ll$ denotes the [[Definition:Element is Way Below|w... | Let $a, b, x, y \in S$ such that
$a \ll x$ and $b \ll y$
By [[Join Succeeds Operands]]:
:$x \preceq x \vee y$ and $y \preceq x \vee y$
By [[Preceding and Way Below implies Way Below]] and definition of [[Definition:Reflexivity|reflexivity]]:
:$a \ll x \vee y$ and $b \ll x \vee y$
Thus by [[Join is Way Below if Opera... | Way Below is Congruent for Join | https://proofwiki.org/wiki/Way_Below_is_Congruent_for_Join | https://proofwiki.org/wiki/Way_Below_is_Congruent_for_Join | [
"Way Below Relation"
] | [
"Definition:Join Semilattice",
"Definition:Congruence Relation",
"Definition:Element is Way Below"
] | [
"Join Succeeds Operands",
"Preceding and Way Below implies Way Below",
"Definition:Reflexivity",
"Join is Way Below if Operands are Way Below"
] |
proofwiki-12372 | Lifting The Exponent Lemma | Let $x, y \in \Z$ be distinct integers.
Let $n \ge 1$ be a natural number.
Let $p$ be an odd prime.
Let:
:$p \divides x - y$
and:
:$p \nmid x y$
where $\divides$ and $\nmid$ denote divisibility and non-divisibility respectively.
Then
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y} + \map {\nu_p} n$
where $\nu_p$ deno... | === Lemma ===
{{:Lifting The Exponent Lemma/Lemma}}{{qed|lemma}}
Let $k = \map {\nu_p} n$.
Then $n = p^k m$ such that $p \nmid m$.
By $p$-adic Valuation of Difference of Powers with Coprime Exponent:
:$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x^{p^k} - y^{p^k} }$
By repeatedly applying the {{Lemma|Lifting The Exponent ... | Let $x, y \in \Z$ be [[Definition:Distinct|distinct]] [[Definition:Integer|integers]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let:
:$p \divides x - y$
and:
:$p \nmid x y$
where $\divides$ and $\nmid$ denote [[Definition:Divisor of Integer|di... | === [[Lifting The Exponent Lemma/Lemma|Lemma]] ===
{{:Lifting The Exponent Lemma/Lemma}}{{qed|lemma}}
Let $k = \map {\nu_p} n$.
Then $n = p^k m$ such that $p \nmid m$.
By [[P-adic Valuation of Difference of Powers with Coprime Exponent|$p$-adic Valuation of Difference of Powers with Coprime Exponent]]:
:$\map {\nu_... | Lifting The Exponent Lemma | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma | [
"P-adic Valuations",
"Number Theory",
"Lifting The Exponent Lemma"
] | [
"Definition:Distinct",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Odd Prime",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:P-adic Valuation"
] | [
"Lifting The Exponent Lemma/Lemma",
"P-adic Valuation of Difference of Powers with Coprime Exponent"
] |
proofwiki-12373 | Zsigmondy's Theorem | Let $a > b > 0$ be coprime positive integers.
Let $n \ge 1$ be a (strictly) positive integer.
Then there is a prime number $p$ such that
:$p$ divides $a^n - b^n$
:$p$ does not divide $a^k - b^k$ for all $k < n$
with the following exceptions:
:$n = 1$ and $a - b = 1$
:$n = 2$ and $a + b$ is a power of $2$
:$n = 6$, $a =... | We call a prime number '''primitive''' if it divides $a^n - b^n$ but not $a^k - b^k$ for any $k < n$.
Let $\map {\Phi_n} {x, y}$ denote the $n$th homogeneous cyclotomic polynomial.
By Product of Cyclotomic Polynomials:
:$a^n - b^n = \ds \prod_{d \mathop \divides n} \map {\Phi_d} {a, b}$
Thus any primitive prime divisor... | Let $a > b > 0$ be [[Definition:Coprime Integers|coprime]] [[Definition:Positive Integer|positive integers]].
Let $n \ge 1$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then there is a [[Definition:Prime Number|prime number]] $p$ such that
:$p$ [[Definition:Divisor of Integer|divides]] ... | We call a [[Definition:Prime Number|prime number]] '''primitive''' if it [[Definition:Divisor of Integer|divides]] $a^n - b^n$ but not $a^k - b^k$ for any $k < n$.
Let $\map {\Phi_n} {x, y}$ denote the $n$th [[Definition:Homogeneous Cyclotomic Polynomial|homogeneous cyclotomic polynomial]].
By [[Product of Cyclotomic... | Zsigmondy's Theorem | https://proofwiki.org/wiki/Zsigmondy's_Theorem | https://proofwiki.org/wiki/Zsigmondy's_Theorem | [
"Number Theory",
"Cyclotomic Polynomials",
"63"
] | [
"Definition:Coprime/Integers",
"Definition:Positive/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Homogeneous Cyclotomic Polynomial",
"Product of Cyclotomic Polynomials",
"Definition:Divisor (Algebra)/Integer",
"Product of Cyclotomic Polynomials"
] |
proofwiki-12374 | Compact Subset is Join Subsemilattice | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $\map K L$ be a compact subset of $L$.
Then $\map K L$ is join subsemilattice:
:$\forall x, y \in \map K L: x \vee y \in \map K L$ | Let $x, y \in \map K L$.
By definition of compact subset:
:$x$ and $y$ are compact.
By definition of compact:
:$x \ll x$ and $y \ll y$
By Way Below is Congruent for Join:
:$x \vee y \ll x \vee y$
By definition:
:$x \vee y$ is compact.
Thus by definition compact subset:
:$x \vee y \in \map K L$
{{qed}} | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $\map K L$ be a [[Definition:Compact Subset of Lattice|compact subset]] of $L$.
Then $\map K L$ is join subsemilattice:
:$\forall x, y \in \map K L: x \vee y \in \map K L$ | Let $x, y \in \map K L$.
By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$x$ and $y$ are [[Definition:Compact Element|compact]].
By definition of [[Definition:Compact Element|compact]]:
:$x \ll x$ and $y \ll y$
By [[Way Below is Congruent for Join]]:
:$x \vee y \ll x \vee y$
By definition... | Compact Subset is Join Subsemilattice | https://proofwiki.org/wiki/Compact_Subset_is_Join_Subsemilattice | https://proofwiki.org/wiki/Compact_Subset_is_Join_Subsemilattice | [
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Compact Subset of Lattice"
] | [
"Definition:Compact Subset of Lattice",
"Definition:Compact Element",
"Definition:Compact Element",
"Way Below is Congruent for Join",
"Definition:Compact Element",
"Definition:Compact Subset of Lattice"
] |
proofwiki-12375 | Totally Ordered Ring Zero Precedes Element or its Inverse | Let $\struct {R, +, \circ, \preceq}$ be an ordered ring.
From the definition of ordered ring, $\preceq$ is compatible with $+$.
Let $0_R$ be the zero element of $R$.
Let $x \ne 0_R$ be a non-zero element of $R$.
Let $-x$ be the ring negative of $x$.
Then:
:$0_R \prec x \lor 0_R \prec -x$
but not both. | By the definition of total ordering, $\preceq$ is connected.
As $x \ne 0_R$, one of the following is true, but not both:
:$(1): \quad 0_R \prec x$
:$(2): \quad x \prec 0_R$
If $(2)$, because $\prec$ is compatible with $+$:
{{begin-eqn}}
{{eqn | l = x
| o = \prec
| r = 0_R
}}
{{eqn | ll= \leadsto
| l ... | Let $\struct {R, +, \circ, \preceq}$ be an [[Definition:Ordered Ring|ordered ring]].
From the definition of [[Definition:Ordered Ring|ordered ring]], $\preceq$ is [[Definition:Relation Compatible with Operation|compatible]] with $+$.
Let $0_R$ be the [[Definition:Ring Zero|zero element]] of $R$.
Let $x \ne 0_R$ be a... | By the definition of [[Definition:Total Ordering|total ordering]], $\preceq$ is [[Definition:Connected Relation|connected]].
As $x \ne 0_R$, one of the following is true, but not both:
:$(1): \quad 0_R \prec x$
:$(2): \quad x \prec 0_R$
If $(2)$, because $\prec$ is [[Definition:Relation Compatible with Operation|c... | Totally Ordered Ring Zero Precedes Element or its Inverse | https://proofwiki.org/wiki/Totally_Ordered_Ring_Zero_Precedes_Element_or_its_Inverse | https://proofwiki.org/wiki/Totally_Ordered_Ring_Zero_Precedes_Element_or_its_Inverse | [
"Ordered Rings"
] | [
"Definition:Ordered Ring",
"Definition:Ordered Ring",
"Definition:Relation Compatible with Operation",
"Definition:Ring Zero",
"Definition:Ring Zero",
"Definition:Ring Negative"
] | [
"Definition:Total Ordering",
"Definition:Connected Relation",
"Definition:Relation Compatible with Operation",
"Category:Ordered Rings"
] |
proofwiki-12376 | Bottom in Compact Subset | Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
Then $\bot \in K\left({L}\right)$
where $\bot$ denotes the smallest element of $L$,
:$K\left({L}\right)$ denotes the compact subset of $L$. | By Bottom is Way Below Any Element:
:$\bot \ll \bot$
where $\ll$ is the way below relation.
By definition:
:$\bot$ is compact.
Thus by definition of compact subset:
:$\bot \in K\left({L}\right)$
{{qed}} | Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Then $\bot \in K\left({L}\right)$
where $\bot$ denotes the [[Definition:Smallest Element|smallest element]] of $L$,
:$K\left({L}\right)$ denotes the [[Definition:Compact Subs... | By [[Bottom is Way Below Any Element]]:
:$\bot \ll \bot$
where $\ll$ is the [[Definition:Element is Way Below|way below relation]].
By definition:
:$\bot$ is [[Definition:Compact Element|compact]].
Thus by definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$\bot \in K\left({L}\right)$
{{qed}} | Bottom in Compact Subset | https://proofwiki.org/wiki/Bottom_in_Compact_Subset | https://proofwiki.org/wiki/Bottom_in_Compact_Subset | [
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Smallest Element",
"Definition:Compact Subset of Lattice"
] | [
"Bottom is Way Below Any Element",
"Definition:Element is Way Below",
"Definition:Compact Element",
"Definition:Compact Subset of Lattice"
] |
proofwiki-12377 | Composition of Relations is not Commutative | Composition of relations is, in general, not commutative.
That is, it is usually the case that:
:$\RR_1 \circ \RR_2 \ne \RR_2 \circ \RR_1$
for relations $\RR_1$ and $\RR_2$. | Proof by Counterexample:
Let $\RR_1 := \struct {S, S, R_1}$ and $\RR_2 := \struct {S, S, R_2}$ be relations defined as:
Let:
:$S = \set {0, 1, 2}$
:$R_1 = \set {\tuple {0, 1} }$
:$R_2 = \set {\tuple {1, 2} }$
We have that:
:$\RR_1 \circ \RR_2 = \struct {S, S, \set {\tuple {0, 2} } }$
while:
:$\RR_2 \circ \RR_1 = \struc... | [[Definition:Composition of Relations|Composition]] of [[Definition:Relation|relations]] is, in general, not [[Definition:Commutative Operation|commutative]].
That is, it is usually the case that:
:$\RR_1 \circ \RR_2 \ne \RR_2 \circ \RR_1$
for [[Definition:Relation|relations]] $\RR_1$ and $\RR_2$. | [[Proof by Counterexample]]:
Let $\RR_1 := \struct {S, S, R_1}$ and $\RR_2 := \struct {S, S, R_2}$ be [[Definition:Relation|relations]] defined as:
Let:
:$S = \set {0, 1, 2}$
:$R_1 = \set {\tuple {0, 1} }$
:$R_2 = \set {\tuple {1, 2} }$
We have that:
:$\RR_1 \circ \RR_2 = \struct {S, S, \set {\tuple {0, 2} } }$
whil... | Composition of Relations is not Commutative | https://proofwiki.org/wiki/Composition_of_Relations_is_not_Commutative | https://proofwiki.org/wiki/Composition_of_Relations_is_not_Commutative | [
"Relation Theory"
] | [
"Definition:Composition of Relations",
"Definition:Relation",
"Definition:Commutative/Operation",
"Definition:Relation"
] | [
"Proof by Counterexample",
"Definition:Relation"
] |
proofwiki-12378 | Compact Closure is Intersection of Lower Closure and Compact Subset | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $x \in S$.
Then $x^{\mathrm {compact} } = x^\preceq \cap \map K L$
where
:$x^{\mathrm {compact} }$ denotes the compact closure of $x$,
:$x^\preceq$ denotes the lower closure of $x$,
:$\map K L$ denotes the compact subset of $L$. | :$y \in x^{\mathrm {compact} }$
{{iff}}
:$y \preceq x$ and $y$ is compact by definition of compact closure
{{iff}}
:$y \in x^\preceq$ and $y$ is compact by definition of lower closure of element
{{iff}}
:$y \in x^\preceq$ and $y \in \map K L$ by definition of compact subset
{{iff}}
:$y \in x^\preceq \cap \map K L$ by d... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $x \in S$.
Then $x^{\mathrm {compact} } = x^\preceq \cap \map K L$
where
:$x^{\mathrm {compact} }$ denotes the [[Definition:Compact Closure|compact closure]] of $x$,
:$x^\pr... | :$y \in x^{\mathrm {compact} }$
{{iff}}
:$y \preceq x$ and $y$ is [[Definition:Compact Element|compact]] by definition of [[Definition:Compact Closure|compact closure]]
{{iff}}
:$y \in x^\preceq$ and $y$ is [[Definition:Compact Element|compact]] by definition of [[Definition:Lower Closure of Element|lower closure of el... | Compact Closure is Intersection of Lower Closure and Compact Subset | https://proofwiki.org/wiki/Compact_Closure_is_Intersection_of_Lower_Closure_and_Compact_Subset | https://proofwiki.org/wiki/Compact_Closure_is_Intersection_of_Lower_Closure_and_Compact_Subset | [
"Way Below Relation",
"Lower Closures"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Compact Closure",
"Definition:Lower Closure/Element",
"Definition:Compact Subset of Lattice"
] | [
"Definition:Compact Element",
"Definition:Compact Closure",
"Definition:Compact Element",
"Definition:Lower Closure/Element",
"Definition:Compact Subset of Lattice",
"Definition:Set Intersection",
"Definition:Set Equality"
] |
proofwiki-12379 | Compact Closure is Subset of Way Below Closure | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then $x^{\mathrm {compact} } \subseteq x^\ll$
where
:$x^{\mathrm {compact} }$ denotes the compact closure of $x$,
:$x^\ll$ denotes the way below closure of $x$. | Let $y \in x^{\mathrm {compact} }$.
By definition of compact closure:
:$y \preceq x$ and $y$ is compact.
By definition of compact:
:$y \ll y$
where $\ll$ denotes the way below relation.
By Preceding and Way Below implies Way Below and definition of reflexivity:
:$y \ll x$
Thus by definition of way below closure:
:$y \i... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then $x^{\mathrm {compact} } \subseteq x^\ll$
where
:$x^{\mathrm {compact} }$ denotes the [[Definition:Compact Closure|compact closure]] of $x$,
:$x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$. | Let $y \in x^{\mathrm {compact} }$.
By definition of [[Definition:Compact Closure|compact closure]]:
:$y \preceq x$ and $y$ is [[Definition:Compact Element|compact]].
By definition of [[Definition:Compact Element|compact]]:
:$y \ll y$
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
By... | Compact Closure is Subset of Way Below Closure | https://proofwiki.org/wiki/Compact_Closure_is_Subset_of_Way_Below_Closure | https://proofwiki.org/wiki/Compact_Closure_is_Subset_of_Way_Below_Closure | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Compact Closure",
"Definition:Way Below Closure"
] | [
"Definition:Compact Closure",
"Definition:Compact Element",
"Definition:Compact Element",
"Definition:Element is Way Below",
"Preceding and Way Below implies Way Below",
"Definition:Reflexivity",
"Definition:Way Below Closure"
] |
proofwiki-12380 | Algebraic iff Continuous and For Every Way Below Exists Compact Between | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice.
Then $L$ is algebraic {{iff}}:
:$L$ is continuous
and:
:$\forall x, y \in S: x \ll y \implies \exists k \in \map K L: x \preceq k \preceq y$
where
:$\ll$ denotes the way below relation,
:$\map K L$ denotes the compact subset of $L$. | === Sufficient Condition ===
Let $L$ be algebraic.
We will prove that
:$\forall x \in S: x^\ll$ is directed
where $x^\ll$ denotes way below closure of $x$.
Let $x \in S$.
By definition of algebraic:
:$x^{\mathrm{compact} }$ is directed
where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.
By Compact Closure... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Then $L$ is [[Definition:Algebraic Ordered Set|algebraic]] {{iff}}:
:$L$ is [[Definition:Continuous Ordered Set|continuous]]
and:
:$\forall x, y \in S: x \ll y \implies \exists k \in \map K L: x \preceq k \preceq y$
where... | === Sufficient Condition ===
Let $L$ be [[Definition:Algebraic Ordered Set|algebraic]].
We will prove that
:$\forall x \in S: x^\ll$ is [[Definition:Directed Subset|directed]]
where $x^\ll$ denotes [[Definition:Way Below Closure|way below closure]] of $x$.
Let $x \in S$.
By definition of [[Definition:Algebraic Orde... | Algebraic iff Continuous and For Every Way Below Exists Compact Between | https://proofwiki.org/wiki/Algebraic_iff_Continuous_and_For_Every_Way_Below_Exists_Compact_Between | https://proofwiki.org/wiki/Algebraic_iff_Continuous_and_For_Every_Way_Below_Exists_Compact_Between | [
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Algebraic Ordered Set",
"Definition:Continuous Ordered Set",
"Definition:Element is Way Below",
"Definition:Compact Subset of Lattice"
] | [
"Definition:Algebraic Ordered Set",
"Definition:Directed Subset",
"Definition:Way Below Closure",
"Definition:Algebraic Ordered Set",
"Definition:Directed Subset",
"Definition:Compact Closure",
"Compact Closure is Subset of Way Below Closure",
"Definition:Non-Empty Set",
"Definition:Subset",
"Non-... |
proofwiki-12381 | Non-Empty Way Below Closure is Directed in Join Semilattice | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $x \in S$ such that:
:$x^\ll \ne \O$
where $x^\ll$ denotes the way below closure of $x$.
Then $x^\ll$ is directed. | By assumption:
:$x^\ll$ is a non-empty set.
Let $y, z \in x^\ll$.
By definition of way below closure:
:$y \ll x$ and $z \ll x$
By Join is Way Below if Operands are Way Below:
:$y \vee z \ll x$
By definition of way below closure:
:$y \vee z \in x^\ll$
By Join Succeeds Operands:
:$y \preceq y \vee z$ and $z \preceq y \ve... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $x \in S$ such that:
:$x^\ll \ne \O$
where $x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$.
Then $x^\ll$ is [[Definition:Directed Subset|directed]]. | By assumption:
:$x^\ll$ is a [[Definition:Non-Empty Set|non-empty set]].
Let $y, z \in x^\ll$.
By definition of [[Definition:Way Below Closure|way below closure]]:
:$y \ll x$ and $z \ll x$
By [[Join is Way Below if Operands are Way Below]]:
:$y \vee z \ll x$
By definition of [[Definition:Way Below Closure|way below... | Non-Empty Way Below Closure is Directed in Join Semilattice | https://proofwiki.org/wiki/Non-Empty_Way_Below_Closure_is_Directed_in_Join_Semilattice | https://proofwiki.org/wiki/Non-Empty_Way_Below_Closure_is_Directed_in_Join_Semilattice | [
"Join and Meet Semilattices",
"Way Below Relation"
] | [
"Definition:Join Semilattice",
"Definition:Way Below Closure",
"Definition:Directed Subset"
] | [
"Definition:Non-Empty Set",
"Definition:Way Below Closure",
"Join is Way Below if Operands are Way Below",
"Definition:Way Below Closure",
"Join Succeeds Operands",
"Definition:Directed Subset"
] |
proofwiki-12382 | Image under Subset of Relation is Subset of Image under Relation | Let $S$ and $T$ be sets.
Let $\RR_1 \subseteq S \times T$ be a relation in $S \times T$.
Let $\RR_2 \subseteq \RR_1$.
Let $A \subseteq S$.
Then:
:$\RR_2 \sqbrk A \subseteq \RR_1 \sqbrk A$
where $\RR_1 \sqbrk A$ denotes the image of $A$ under $\RR_1$. | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR_2 \sqbrk A
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in A
| l = \tuple {x, y}
| o = \in
| r = \RR_2
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | ll= \leadsto
| q = \exists x \in A
| l = \tuple ... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR_1 \subseteq S \times T$ be a [[Definition:Relation|relation]] in $S \times T$.
Let $\RR_2 \subseteq \RR_1$.
Let $A \subseteq S$.
Then:
:$\RR_2 \sqbrk A \subseteq \RR_1 \sqbrk A$
where $\RR_1 \sqbrk A$ denotes the [[Definition:Image of Subset under Relation|imag... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR_2 \sqbrk A
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in A
| l = \tuple {x, y}
| o = \in
| r = \RR_2
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | ll= \leadsto
| q = \exists x \in A
| l = \tuple ... | Image under Subset of Relation is Subset of Image under Relation | https://proofwiki.org/wiki/Image_under_Subset_of_Relation_is_Subset_of_Image_under_Relation | https://proofwiki.org/wiki/Image_under_Subset_of_Relation_is_Subset_of_Image_under_Relation | [
"Relation Theory"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Image (Set Theory)/Relation/Subset"
] | [] |
proofwiki-12383 | Intersection of Transitive Relations is Transitive/General Result | Let $\family {\RR_i: i \mathop \in I}$ be an $I$-indexed set of transitive relations on a set $S$.
Then their intersection $\ds \bigcap_{i \mathop \in I} \RR_i$ is also a transitive relation on $S$. | Let $\ds \tuple {x, y} \in \bigcap_{i \mathop \in I} \RR_i$ and $\ds \tuple {y, z} \in \bigcap_{i \mathop \in I} \RR_i$ be transitive relations on an arbitrary set $S$.
Then by definition of intersection:
:$\tuple {x, y} \in \RR_i$ for all $i \in I$
:$\tuple {y, z} \in \RR_i$ for all $i \in I$
Since each $\RR_i$ is tra... | Let $\family {\RR_i: i \mathop \in I}$ be an [[Definition:Indexed Set|$I$-indexed set]] of [[Definition:Transitive Relation|transitive relations]] on a [[Definition:Set|set]] $S$.
Then their [[Definition:Set Intersection|intersection]] $\ds \bigcap_{i \mathop \in I} \RR_i$ is also a [[Definition:Transitive Relation|t... | Let $\ds \tuple {x, y} \in \bigcap_{i \mathop \in I} \RR_i$ and $\ds \tuple {y, z} \in \bigcap_{i \mathop \in I} \RR_i$ be [[Definition:Transitive Relation|transitive relations]] on an arbitrary [[Definition:Set|set]] $S$.
Then by definition of [[Definition:Intersection of Family|intersection]]:
:$\tuple {x, y} \in \... | Intersection of Transitive Relations is Transitive/General Result | https://proofwiki.org/wiki/Intersection_of_Transitive_Relations_is_Transitive/General_Result | https://proofwiki.org/wiki/Intersection_of_Transitive_Relations_is_Transitive/General_Result | [
"Intersection of Transitive Relations is Transitive"
] | [
"Definition:Indexing Set/Indexed Set",
"Definition:Transitive Relation",
"Definition:Set",
"Definition:Set Intersection",
"Definition:Transitive Relation"
] | [
"Definition:Transitive Relation",
"Definition:Set",
"Definition:Set Intersection/Family of Sets",
"Definition:Transitive Relation",
"Definition:Set Intersection/Family of Sets",
"Definition:Transitive Relation",
"Category:Intersection of Transitive Relations is Transitive"
] |
proofwiki-12384 | Upper Bounds are Equivalent implies Suprema are equal | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $X, Y$ be subsets of $S$.
Assume that
:$X$ admits a supremum
and
:$\forall x \in S: x$ is upper bound for $X \iff x$ is upper bound for $Y$
Then $\sup X = \sup Y$ | We will prove that
:$\forall b \in S: b$ is upper bound for $Y \implies \sup X \preceq b$
Let $b \in S$ such that
:$b$ is upper bound for $Y$.
By assumption:
:$b$ is upper bound for $X$.
Thus by definition of supremum:
:$\sup X \preceq b$
{{qed|lemma}}
By definition of supremum:
:$\sup X$ is upper bound for $X$.
By ass... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X, Y$ be [[Definition:Subset|subsets]] of $S$.
Assume that
:$X$ admits a [[Definition:Supremum of Set|supremum]]
and
:$\forall x \in S: x$ is [[Definition:Upper Bound of Set|upper bound]] for $X \iff x$ is [[Definition:Upper Bound of Se... | We will prove that
:$\forall b \in S: b$ is [[Definition:Upper Bound of Set|upper bound]] for $Y \implies \sup X \preceq b$
Let $b \in S$ such that
:$b$ is [[Definition:Upper Bound of Set|upper bound]] for $Y$.
By assumption:
:$b$ is [[Definition:Upper Bound of Set|upper bound]] for $X$.
Thus by definition of [[Defi... | Upper Bounds are Equivalent implies Suprema are equal | https://proofwiki.org/wiki/Upper_Bounds_are_Equivalent_implies_Suprema_are_equal | https://proofwiki.org/wiki/Upper_Bounds_are_Equivalent_implies_Suprema_are_equal | [
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set"
] |
proofwiki-12385 | Inverse of Subset of Relation is Subset of Inverse | Let $S$ and $T$ be sets
Let $\RR_1 = S \times T$ be a relation on $S \times T$.
Let $\RR_2 \subseteq \RR_1$.
Then:
:$\RR_2^{-1} \subseteq \RR_1^{-1}$
where $\RR_1^{-1}$ denotes the inverse of $\RR_1$. | {{begin-eqn}}
{{eqn | l = \tuple {t, s}
| o = \in
| r = \RR_2^{-1}
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {s, t}
| o = \in
| r = \RR_2
| c = {{Defof|Inverse Relation}}
}}
{{eqn | ll= \leadsto
| l = \tuple {s, t}
| o = \in
| r = \RR_1
| c = {{Defof|S... | Let $S$ and $T$ be [[Definition:Set|sets]]
Let $\RR_1 = S \times T$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $\RR_2 \subseteq \RR_1$.
Then:
:$\RR_2^{-1} \subseteq \RR_1^{-1}$
where $\RR_1^{-1}$ denotes the [[Definition:Inverse Relation|inverse]] of $\RR_1$. | {{begin-eqn}}
{{eqn | l = \tuple {t, s}
| o = \in
| r = \RR_2^{-1}
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {s, t}
| o = \in
| r = \RR_2
| c = {{Defof|Inverse Relation}}
}}
{{eqn | ll= \leadsto
| l = \tuple {s, t}
| o = \in
| r = \RR_1
| c = {{Defof|S... | Inverse of Subset of Relation is Subset of Inverse | https://proofwiki.org/wiki/Inverse_of_Subset_of_Relation_is_Subset_of_Inverse | https://proofwiki.org/wiki/Inverse_of_Subset_of_Relation_is_Subset_of_Inverse | [
"Subsets",
"Inverse Relations"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Inverse Relation"
] | [
"Definition:Subset",
"Category:Subsets",
"Category:Inverse Relations"
] |
proofwiki-12386 | Preceding implies if Less Upper Bound then Greater Upper Bound | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$ such that
:$x \preceq y$
Let $X \subseteq S$.
Then
:$x$ is upper bound for $X \implies y$ is upper bound for $X$
and
:$y$ is lower bound for $X \implies x$ is lower bound for $X$. | === First Implication ===
Let $x$ be upper bound for $X$,
Let $z \in X$.
By definition of upper bound:
:$z \preceq x$
Thus by definition of transitivity:
:$z \preceq y$
{{qed|lemma}} | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x \preceq y$
Let $X \subseteq S$.
Then
:$x$ is [[Definition:Upper Bound of Set|upper bound]] for $X \implies y$ is [[Definition:Upper Bound of Set|upper bound]] for $X$
and
:$y$ is [[Definition:Lower Bound of S... | === First Implication ===
Let $x$ be [[Definition:Upper Bound of Set|upper bound]] for $X$,
Let $z \in X$.
By definition of [[Definition:Upper Bound of Set|upper bound]]:
:$z \preceq x$
Thus by definition of [[Definition:Transitivity|transitivity]]:
:$z \preceq y$
{{qed|lemma}} | Preceding implies if Less Upper Bound then Greater Upper Bound | https://proofwiki.org/wiki/Preceding_implies_if_Less_Upper_Bound_then_Greater_Upper_Bound | https://proofwiki.org/wiki/Preceding_implies_if_Less_Upper_Bound_then_Greater_Upper_Bound | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Transitive"
] |
proofwiki-12387 | Three Regular Tessellations/Squares | Squares form a regular tessellation:
:400px | {{proof wanted}}
Category:Three Regular Tessellations
Category:Squares
nq7qhbj876vmocxcyo6nw999ylgysrj | [[Definition:Square (Geometry)|Squares]] form a [[Definition:Regular Tessellation|regular tessellation]]:
:[[File:RegularSquareTessellation.png|400px]] | {{proof wanted}}
[[Category:Three Regular Tessellations]]
[[Category:Squares]]
nq7qhbj876vmocxcyo6nw999ylgysrj | Three Regular Tessellations/Squares | https://proofwiki.org/wiki/Three_Regular_Tessellations/Squares | https://proofwiki.org/wiki/Three_Regular_Tessellations/Squares | [
"Three Regular Tessellations",
"Squares"
] | [
"Definition:Quadrilateral/Square",
"Definition:Regular Tessellation",
"File:RegularSquareTessellation.png"
] | [
"Category:Three Regular Tessellations",
"Category:Squares"
] |
proofwiki-12388 | Three Regular Tessellations/Triangles | Equilateral triangles form a regular tessellation:
:400px | Let $s \in \R_{>0}$ be the side length of the equilateral triangles.
For all $x, y \in \Z$, let the center $P_{x, y}$ of each triangle have Cartesian coordinates:
:$P_{x, y} = s \tuple {\dfrac 1 2 x, \dfrac {\sqrt 3} 2 y + \dfrac {3 - m} {4 \sqrt 3} }$
where $m = \begin{cases} 1 & \textrm {for $x + y$ even} \\ -1 & \te... | [[Definition:Equilateral Triangle|Equilateral triangles]] form a [[Definition:Regular Tessellation|regular tessellation]]:
:[[File:RegularTriangleTessellation.png|400px]] | Let $s \in \R_{>0}$ be the [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] of the [[Definition:Equilateral Triangle|equilateral triangles]].
For all $x, y \in \Z$, let the [[Definition:Center of Regular Polygon|center]] $P_{x, y}$ of each [[Definition:Triangle (Geometry)|triangle]] have [[Defi... | Three Regular Tessellations/Triangles | https://proofwiki.org/wiki/Three_Regular_Tessellations/Triangles | https://proofwiki.org/wiki/Three_Regular_Tessellations/Triangles | [
"Three Regular Tessellations",
"Equilateral Triangles"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Regular Tessellation",
"File:RegularTriangleTessellation.png"
] | [
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Regular/Center",
"Definition:Triangle (Geometry)",
"Definition:Cartesian Plane/Ordered Pair",
"Definition:Triangle (Geometry)",
"Definition:Cartesian Plane/Ordered Pair",
... |
proofwiki-12389 | Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Then:
:$L$ is arithmetic
{{iff}}:
:$\ll$ is a multiplicative relation
where $\ll$ denotes the way below relation of $L$. | === Sufficient Condition ===
Let $L$ be arithmetic.
Let $a, x, y \in S$ such that
:$a \ll x$ and $a \ll y$
By Algebraic iff Continuous and For Every Way Below Exists Compact Between:
:$\exists c \in \map K L: a \preceq c \preceq x$
and
:$\exists k \in \map K L: a \preceq k \preceq y$
where $\map K L$ denotes the compac... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Then:
:$L$ is [[Definition:Arithmetic Ordered Set|arithmetic]]
{{iff}}:
:$\ll$ is a [[Definition:Multiplicative Relation|multip... | === Sufficient Condition ===
Let $L$ be [[Definition:Arithmetic Ordered Set|arithmetic]].
Let $a, x, y \in S$ such that
:$a \ll x$ and $a \ll y$
By [[Algebraic iff Continuous and For Every Way Below Exists Compact Between]]:
:$\exists c \in \map K L: a \preceq c \preceq x$
and
:$\exists k \in \map K L: a \preceq k \... | Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice | https://proofwiki.org/wiki/Arithmetic_iff_Way_Below_Relation_is_Multiplicative_in_Algebraic_Lattice | https://proofwiki.org/wiki/Arithmetic_iff_Way_Below_Relation_is_Multiplicative_in_Algebraic_Lattice | [
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Arithmetic Ordered Set",
"Definition:Multiplicative Relation",
"Definition:Element is Way Below"
] | [
"Definition:Arithmetic Ordered Set",
"Algebraic iff Continuous and For Every Way Below Exists Compact Between",
"Definition:Compact Subset of Lattice",
"Meet Semilattice is Ordered Structure",
"Definition:Arithmetic Ordered Set",
"Definition:Meet Closed",
"Definition:Meet Closed",
"Definition:Compact ... |
proofwiki-12390 | Endomorphism Ring of Abelian Group is Ring with Unity | Let $\struct {G, +}$ be an abelian group.
Let $\struct {\map {\mathrm {End} } G, \oplus, \circ}$ be its endomorphism ring.
Then $\struct {\map {\mathrm {End} } G, \oplus, \circ}$ is a ring with unity $I_G$, where:
:$I_G$ is the identity mapping on $G$
:$\oplus$ is the pointwise operation on $\map {\mathrm {End} } G$ in... | {{tidy|throughout, and correction of grammar}}
{{proofread}}
In the below, $f, g , h \in \map {\mathrm {End} } G$.
As $f, g, h$ are (group) endomorphisms, it follows {{afortiori}} that they are also (group) homomorphisms.
Recall that $f \oplus g$ is the pointwise operation on $f$ and $g$ induced by $+$.
We are given th... | Let $\struct {G, +}$ be an [[Definition:Abelian Group|abelian group]].
Let $\struct {\map {\mathrm {End} } G, \oplus, \circ}$ be its [[Definition:Endomorphism Ring of Abelian Group|endomorphism ring]].
Then $\struct {\map {\mathrm {End} } G, \oplus, \circ}$ is a [[Definition:Ring with Unity|ring]] with [[Definition:... | {{tidy|throughout, and correction of grammar}}
{{proofread}}
In the below, $f, g , h \in \map {\mathrm {End} } G$.
As $f, g, h$ are [[Definition:Group Endomorphism|(group) endomorphisms]], it follows {{afortiori}} that they are also [[Definition:Group Homomorphism|(group) homomorphisms]].
Recall that $f \oplus g$ i... | Endomorphism Ring of Abelian Group is Ring with Unity | https://proofwiki.org/wiki/Endomorphism_Ring_of_Abelian_Group_is_Ring_with_Unity | https://proofwiki.org/wiki/Endomorphism_Ring_of_Abelian_Group_is_Ring_with_Unity | [
"Abstract Algebra",
"Rings of Endomorphisms",
"Endomorphism Rings of Abelian Groups"
] | [
"Definition:Abelian Group",
"Definition:Endomorphism Ring/Abelian Group",
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Identity Mapping",
"Definition:Pointwise Operation"
] | [
"Definition:Group Endomorphism",
"Definition:Group Homomorphism",
"Definition:Pointwise Operation",
"Definition:Pointwise Operation",
"Definition:Given",
"Definition:Abelian Group",
"Definition:Commutative Semigroup",
"Homomorphism on Induced Structure to Commutative Semigroup",
"Definition:Homomorp... |
proofwiki-12391 | Elements of Finite Support form Submagma of Direct Product | Let $\struct {S_i, \circ_i}_{i \mathop \in I}$ be a family of magmas with identity.
Let $\ds S = \prod_{i \mathop \in I} S_i$ be their direct product.
Let $T$ be the subset of elements of $S$ whose support is finite:
:$T = \set {s \in S: \map \supp s \text{ is finite} }$
Then $T$ is a submagma of $S$. | From Finite Subsets form Ideal, the set of finite subsets of $I$ form an ideal of $I$.
From Elements with Support in Ideal form Submagma of Direct Product, $T$ is a submagma of $S$.
Category:Direct Products
0ktxy08bwzt1vfvd8mca0qi51wrsjov | Let $\struct {S_i, \circ_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Magma|magmas]] with [[Definition:Identity Element|identity]].
Let $\ds S = \prod_{i \mathop \in I} S_i$ be their [[Definition:External Direct Product|direct product]].
Let $T$ be the [[Definition:Subset|subset]] o... | From [[Finite Subsets form Ideal]], the set of [[Definition:Finite Subset|finite subsets]] of $I$ form an [[Definition:Ideal (Order Theory)|ideal]] of $I$.
From [[Elements with Support in Ideal form Submagma of Direct Product]], $T$ is a [[Definition:Submagma|submagma]] of $S$.
[[Category:Direct Products]]
0ktxy08bwz... | Elements of Finite Support form Submagma of Direct Product | https://proofwiki.org/wiki/Elements_of_Finite_Support_form_Submagma_of_Direct_Product | https://proofwiki.org/wiki/Elements_of_Finite_Support_form_Submagma_of_Direct_Product | [
"Direct Products"
] | [
"Definition:Indexing Set/Family",
"Definition:Magma",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:External Direct Product",
"Definition:Subset",
"Definition:Element",
"Definition:Support of Element of Direct Product",
"Definition:Finite Set",
"Definition:Submagma"
] | [
"Finite Subsets form Ideal",
"Definition:Finite Subset",
"Definition:Ideal (Order Theory)",
"Elements with Support in Ideal form Submagma of Direct Product",
"Definition:Submagma",
"Category:Direct Products"
] |
proofwiki-12392 | Möbius Inversion Formula/Abelian Group | Let $G$ be an abelian group.
Let $f, g: \N \to G$ be mappings.
Then
:$\ds \map f n = \prod_{d \mathop \divides n} \map g d$
{{iff}}:
:$\ds \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$ | {{ProofWanted|The proof should go in the same way as in the main theorem ($G {{=}} \R$ or $\C$).}}
Category:Möbius Inversion Formula
Category:Abelian Groups
Category:Möbius Function
okdav3r3k2kigfwpo1mhiw4bpdiovx7 | Let $G$ be an [[Definition:Abelian Group|abelian group]].
Let $f, g: \N \to G$ be [[Definition:Mapping|mappings]].
Then
:$\ds \map f n = \prod_{d \mathop \divides n} \map g d$
{{iff}}:
:$\ds \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$ | {{ProofWanted|The proof should go in the same way as in the main theorem ($G {{=}} \R$ or $\C$).}}
[[Category:Möbius Inversion Formula]]
[[Category:Abelian Groups]]
[[Category:Möbius Function]]
okdav3r3k2kigfwpo1mhiw4bpdiovx7 | Möbius Inversion Formula/Abelian Group | https://proofwiki.org/wiki/Möbius_Inversion_Formula/Abelian_Group | https://proofwiki.org/wiki/Möbius_Inversion_Formula/Abelian_Group | [
"Möbius Inversion Formula",
"Abelian Groups",
"Möbius Function"
] | [
"Definition:Abelian Group",
"Definition:Mapping"
] | [
"Category:Möbius Inversion Formula",
"Category:Abelian Groups",
"Category:Möbius Function"
] |
proofwiki-12393 | Möbius Inversion Formula for Cyclotomic Polynomials | Let $n > 0$ be a (strictly) positive integer.
Let $\Phi_n$ be the $n$th cyclotomic polynomial.
Then:
:$\map {\Phi_n} x = \ds \prod_{d \mathop \divides n} \paren {x^d - 1}^{\map \mu {n / d} }$
where:
:the product runs over all divisors of $n$
:$\mu$ is the Möbius function. | By Product of Cyclotomic Polynomials:
:$\ds \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$
for all $n \in \N$.
The nonzero rational forms form an abelian group under multiplication.
By the Möbius inversion formula for abelian groups, this implies:
:$\ds \map {\Phi_n} x = \prod_{d \mathop \divides n} \paren {x^... | Let $n > 0$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\Phi_n$ be the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
Then:
:$\map {\Phi_n} x = \ds \prod_{d \mathop \divides n} \paren {x^d - 1}^{\map \mu {n / d} }$
where:
:the product runs over all [[Definition... | By [[Product of Cyclotomic Polynomials]]:
:$\ds \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$
for all $n \in \N$.
The nonzero [[Definition:Rational Form|rational forms]] form an [[Definition:Abelian Group|abelian group]] under multiplication.
By the [[Möbius Inversion Formula/Abelian Group|Möbius inversio... | Möbius Inversion Formula for Cyclotomic Polynomials | https://proofwiki.org/wiki/Möbius_Inversion_Formula_for_Cyclotomic_Polynomials | https://proofwiki.org/wiki/Möbius_Inversion_Formula_for_Cyclotomic_Polynomials | [
"Cyclotomic Polynomials"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Cyclotomic Polynomial",
"Definition:Divisor (Algebra)/Integer",
"Definition:Möbius Function"
] | [
"Product of Cyclotomic Polynomials",
"Definition:Rational Form",
"Definition:Abelian Group",
"Möbius Inversion Formula/Abelian Group",
"Category:Cyclotomic Polynomials"
] |
proofwiki-12394 | Composite of Inverse of Mapping with Mapping | Let $f: S \to T$ be a mapping.
Then:
:$f \circ f^{-1} = I_{\Img f}$
where:
: $f \circ f^{-1}$ is the composite of $f$ and $f^{-1}$
: $f^{-1}$ is the inverse of $f$
: $I_{\Img f}$ is the identity mapping on the image set of $f$. | By Inverse of Mapping is One-to-Many Relation, $f^{-1}$ is a one-to-many relation:
:$f^{-1} \subseteq T \times S$
whose domain is the image set of $f$.
By definition of composition of relations:
:$f \circ f^{-1} := \set {\tuple {x, z} \in T \times T: \exists y \in S: \tuple {x, y} \in f^{-1} \land \tuple {y, z} \in f}$... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$f \circ f^{-1} = I_{\Img f}$
where:
: $f \circ f^{-1}$ is the [[Definition:Composition of Relations|composite]] of $f$ and $f^{-1}$
: $f^{-1}$ is the [[Definition:Inverse of Mapping|inverse]] of $f$
: $I_{\Img f}$ is the [[Definition:Identity Mapping|ident... | By [[Inverse of Mapping is One-to-Many Relation]], $f^{-1}$ is a [[Definition:One-to-Many Relation|one-to-many relation]]:
:$f^{-1} \subseteq T \times S$
whose [[Definition:Domain of Relation|domain]] is the [[Definition:Image of Mapping|image set of $f$]].
By definition of [[Definition:Composition of Relations|compos... | Composite of Inverse of Mapping with Mapping | https://proofwiki.org/wiki/Composite_of_Inverse_of_Mapping_with_Mapping | https://proofwiki.org/wiki/Composite_of_Inverse_of_Mapping_with_Mapping | [
"Equivalence Relations",
"Inverse Mappings",
"Composite Mappings"
] | [
"Definition:Mapping",
"Definition:Composition of Relations",
"Definition:Inverse of Mapping",
"Definition:Identity Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Inverse of Mapping is One-to-Many Relation",
"Definition:One-to-Many Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Composition of Relations",
"Definition:Relation",
"Definition:Mapping",
"Definition:Many-to-One Relation",
"Definit... |
proofwiki-12395 | Equivalence of Definitions of Order Complete Set | Let $\struct {S, \preceq}$ be an ordered set.
{{TFAE|def = Order Complete Set}} | === Definition 1 implies Definition 2 ===
Let $\struct {S, \preceq}$ be an order complete set by definition 1.
Let $H \subseteq S$ have a lower bound.
Let $K$ be the set of all lower bounds of $H$.
Then $K \ne \O$.
By definition of lower bound:
:$\forall x \in K: \forall y \in H: x \le y$
and so all elements of $H$ are... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
{{TFAE|def = Order Complete Set}} | === Definition 1 implies Definition 2 ===
Let $\struct {S, \preceq}$ be an [[Definition:Order Complete Set/Definition 1|order complete set by definition 1]].
Let $H \subseteq S$ have a [[Definition:Lower Bound of Set|lower bound]].
Let $K$ be the [[Definition:Set|set]] of all [[Definition:Lower Bound of Set|lower bo... | Equivalence of Definitions of Order Complete Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_Complete_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_Complete_Set | [
"Order Theory"
] | [
"Definition:Ordered Set"
] | [
"Definition:Order Complete Set/Definition 1",
"Definition:Lower Bound of Set",
"Definition:Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Element",
"Definition:Upper Bound of Set",
"Definition:By Hypothesis",
"Definition:Supremum",
"Definition:Supremum of Set",... |
proofwiki-12396 | Condition for Uniqueness of Increasing Mappings between Tosets | Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets.
Let $f: S \to T$ and $g: S \to T$ be increasing mappings from $S$ to $T$.
Let $H \subseteq S$ be a subset of $S$.
Let $f$ and $g$ agree on $H$.
Let $K = f \sqbrk H$ be the image set of $H$ under $f$.
Let the intersection of $K$ with every set of the ... | By hypothesis, let the intersection of $K$ with every set of the form:
:$\set {y \in T: u \prec y \prec v: u, v \in T, u \prec v}$
be non-empty.
{{AimForCont}} $f \ne g$.
Then:
:$\exists x \in S: \map f x \ne \map g x$
{{WLOG}}, suppose that $\map f x < \map g x$.
Let:
:$a \in H$ such that $a \preceq x$
:$b \in H$ such... | Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be [[Definition:Totally Ordered Set|tosets]].
Let $f: S \to T$ and $g: S \to T$ be [[Definition:Increasing Mapping|increasing mappings]] from $S$ to $T$.
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $f$ and $g$ [[Definition:Agreement of ... | [[Definition:By Hypothesis|By hypothesis]], let the [[Definition:Set Intersection|intersection]] of $K$ with every [[Definition:Set|set]] of the form:
:$\set {y \in T: u \prec y \prec v: u, v \in T, u \prec v}$
be [[Definition:Non-Empty Set|non-empty]].
{{AimForCont}} $f \ne g$.
Then:
:$\exists x \in S: \map f x \ne... | Condition for Uniqueness of Increasing Mappings between Tosets | https://proofwiki.org/wiki/Condition_for_Uniqueness_of_Increasing_Mappings_between_Tosets | https://proofwiki.org/wiki/Condition_for_Uniqueness_of_Increasing_Mappings_between_Tosets | [
"Total Orderings"
] | [
"Definition:Totally Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Subset",
"Definition:Agreement/Mappings",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Set Intersection",
"Definition:Set",
"Definition:Non-Empty Set"
] | [
"Definition:By Hypothesis",
"Definition:Set Intersection",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Increasing/Mapping",
"Definition:Element",
"Definition:Set",
"Definition:Set Intersection",
"Definition:Set",
"Definition:Empty Set",
"Definition:Contradiction",
"Proof by Contr... |
proofwiki-12397 | Characteristic of Increasing Mapping from Toset to Order Complete Toset | Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets.
Let $T$ be order complete.
Let $H \subseteq S$ be a subset of $S$.
Let $f: H \to T$ be an increasing mapping from $H$ to $T$.
Then:
:$f$ has an extension to $S$ which is increasing
{{iff}}:
:for all $A \subseteq H$: if $A$ is bounded in $S$, then $f ... | === Necessary Condition ===
Suppose $f$ is such that it is not the case that:
:for all $A \subseteq H$: if $A$ is bounded in $S$, then $f \sqbrk A$ is bounded in $T$.
Proof by Counterexample:
Let $S = \R_{>0}$ be the set of all (strictly) positive real numbers.
Let $H \subseteq S$ be the open real interval $H = \openin... | Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be [[Definition:Totally Ordered Set|tosets]].
Let $T$ be [[Definition:Order Complete Set|order complete]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $f: H \to T$ be an [[Definition:Increasing Mapping|increasing mapping]] from $H$ to $... | === Necessary Condition ===
Suppose $f$ is such that it is not the case that:
:for all $A \subseteq H$: if $A$ is [[Definition:Bounded Set|bounded in $S$]], then $f \sqbrk A$ is [[Definition:Bounded Set|bounded in $T$]].
[[Proof by Counterexample]]:
Let $S = \R_{>0}$ be the [[Definition:Set|set]] of all [[Definition... | Characteristic of Increasing Mapping from Toset to Order Complete Toset | https://proofwiki.org/wiki/Characteristic_of_Increasing_Mapping_from_Toset_to_Order_Complete_Toset | https://proofwiki.org/wiki/Characteristic_of_Increasing_Mapping_from_Toset_to_Order_Complete_Toset | [
"Bounded Sets",
"Total Orderings",
"Order Complete Sets"
] | [
"Definition:Totally Ordered Set",
"Definition:Order Complete Set",
"Definition:Subset",
"Definition:Increasing/Mapping",
"Definition:Extension of Mapping",
"Definition:Increasing/Mapping",
"Definition:Bounded Set",
"Definition:Bounded Set",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Definition:Bounded Set",
"Definition:Bounded Set",
"Proof by Counterexample",
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Real Interval/Open",
"Definition:Real Number/Axiomatic Definition",
"Definition:Totally Ordered Set",
"Continuum Property",
"Definition:Order Com... |
proofwiki-12398 | Arithmetic iff Compact Subset form Lattice in Algebraic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Then:
:$L$ is arithmetic
{{iff}}:
:$\struct {\map K L, \precsim}$ is a lattice
where $\map K L$ denotes the compact subset of $L$:
:$\mathord \precsim = \mathord \preceq \cap \paren {\map K L \times \map K L}$ | === Sufficient Condition ===
Define $K = \struct {\map K L, \precsim}$.
Let $x, y \in \map K L$.
By Compact Subset is Join Subsemilattice:
:$x \vee_L y \in \map K L$
We will prove that
:$x \vee_L y$ is upper bound for $\set {x, y}$ in $K$
Let $z \in \set {x, y}$.
By Join Succeeds Operands:
:$z \preceq x \vee_L y$
By de... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Then:
:$L$ is [[Definition:Arithmetic Ordered Set|arithmetic]]
{{iff}}:
:$\struct {\map K L, \precsim}$ is a [[Definition:Latti... | === Sufficient Condition ===
Define $K = \struct {\map K L, \precsim}$.
Let $x, y \in \map K L$.
By [[Compact Subset is Join Subsemilattice]]:
:$x \vee_L y \in \map K L$
We will prove that
:$x \vee_L y$ is [[Definition:Upper Bound of Set|upper bound]] for $\set {x, y}$ in $K$
Let $z \in \set {x, y}$.
By [[Join Su... | Arithmetic iff Compact Subset form Lattice in Algebraic Lattice | https://proofwiki.org/wiki/Arithmetic_iff_Compact_Subset_form_Lattice_in_Algebraic_Lattice | https://proofwiki.org/wiki/Arithmetic_iff_Compact_Subset_form_Lattice_in_Algebraic_Lattice | [
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Arithmetic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Compact Subset of Lattice"
] | [
"Compact Subset is Join Subsemilattice",
"Definition:Upper Bound of Set",
"Join Succeeds Operands",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definiti... |
proofwiki-12399 | Four Kepler-Poinsot Polyhedra | There exist exactly four Kepler-Poinsot polyhedra:
{{begin-itemize}}
{{item|(1):|the small stellated dodecahedron}}
{{item|(2):|the great stellated dodecahedron}}
{{item|(3):|the great dodecahedron}}
{{item|(4):|the great icosahedron.}}
{{end-itemize}} | {{ProofWanted|needs considerable vision}} | There exist exactly four [[Definition:Kepler-Poinsot Polyhedron|Kepler-Poinsot polyhedra]]:
{{begin-itemize}}
{{item|(1):|the [[Definition:Small Stellated Dodecahedron|small stellated dodecahedron]]}}
{{item|(2):|the [[Definition:Great Stellated Dodecahedron|great stellated dodecahedron]]}}
{{item|(3):|the [[Definition... | {{ProofWanted|needs considerable vision}} | Four Kepler-Poinsot Polyhedra | https://proofwiki.org/wiki/Four_Kepler-Poinsot_Polyhedra | https://proofwiki.org/wiki/Four_Kepler-Poinsot_Polyhedra | [
"Kepler-Poinsot Polyhedra",
"4"
] | [
"Definition:Kepler-Poinsot Polyhedron",
"Definition:Small Stellated Dodecahedron",
"Definition:Great Stellated Dodecahedron",
"Definition:Great Dodecahedron",
"Definition:Great Icosahedron"
] | [] |
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