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-1900 | Closed Topologist's Sine Curve is not Path-Connected | Let $T$ be the closed topologist's sine curve.
Then $T$ is '''not''' path-connected. | {{Recall|Closed Topologist's Sine Curve}}
{{:Definition:Closed Topologist's Sine Curve}}
Let $I = \closedint 0 1 \subseteq \R$ be the closed real interval.
Let $A = \tuple {\dfrac 1 \pi, 0} \in \R^2$.
This proof is based on the fact that a continuous path $f: I \to G \cup J$ beginning at $A$ will never actually arrive ... | Let $T$ be the [[Definition:Closed Topologist's Sine Curve|closed topologist's sine curve]].
Then $T$ is '''not''' [[Definition:Path-Connected Space|path-connected]]. | {{Recall|Closed Topologist's Sine Curve}}
{{:Definition:Closed Topologist's Sine Curve}}
Let $I = \closedint 0 1 \subseteq \R$ be the [[Definition:Closed Real Interval|closed real interval]].
Let $A = \tuple {\dfrac 1 \pi, 0} \in \R^2$.
This proof is based on the fact that a [[Definition:Path (Topology)|continuous p... | Closed Topologist's Sine Curve is not Path-Connected | https://proofwiki.org/wiki/Closed_Topologist's_Sine_Curve_is_not_Path-Connected | https://proofwiki.org/wiki/Closed_Topologist's_Sine_Curve_is_not_Path-Connected | [
"Closed Topologist's Sine Curve",
"Examples of Path-Connected Spaces"
] | [
"Definition:Closed Topologist's Sine Curve",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Real Interval/Closed",
"Definition:Path (Topology)",
"Definition:Compact Space/Real Analysis",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Inclusion Mapping",
"Definition:Projection (Mapping Theory)",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition... |
proofwiki-1901 | Sum of Two Sides of Triangle Greater than Third Side | Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.
{{:Euclid:Proposition/I/20}} | :350 px
Let $ABC$ be a triangle.
By {{EuclidPostulateLink|Second}}, we can produce $BA$ past $A$ in a straight line.
By Construction of Equal Straight Lines from Unequal, there exists a point $D$ such that $DA = CA$.
Therefore, from Isosceles Triangle has Two Equal Angles:
:$\angle ADC = \angle ACD$
Thus by {{EuclidCom... | Given a [[Definition:Triangle (Geometry)|triangle]] $ABC$, the sum of the [[Definition:Length of Line|lengths]] of any two [[Definition:Side of Polygon|sides]] of the triangle is greater than the [[Definition:Length of Line|length]] of the third [[Definition:Side of Polygon|side]].
{{:Euclid:Proposition/I/20}} | :[[File:Euclid-I-20.png|350 px]]
Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
By {{EuclidPostulateLink|Second}}, we can [[Definition:Production|produce]] $BA$ past $A$ in a [[Definition:Straight Line|straight line]].
By [[Construction of Equal Straight Lines from Unequal]], there exists a [[Definition... | Sum of Two Sides of Triangle Greater than Third Side | https://proofwiki.org/wiki/Sum_of_Two_Sides_of_Triangle_Greater_than_Third_Side | https://proofwiki.org/wiki/Sum_of_Two_Sides_of_Triangle_Greater_than_Third_Side | [
"Triangles",
"Triangle Inequality"
] | [
"Definition:Triangle (Geometry)",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side"
] | [
"File:Euclid-I-20.png",
"Definition:Triangle (Geometry)",
"Definition:Production",
"Definition:Line/Straight Line",
"Construction of Equal Straight Lines from Unequal",
"Definition:Point",
"Isosceles Triangle has Two Equal Angles",
"Definition:Triangle (Geometry)",
"Greater Angle of Triangle Subtend... |
proofwiki-1902 | Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides | Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle.
{{:Euclid:Proposition/I/21}} | :250px
Given a triangle $ABC$ and a point $D$ inside it.
We can construct lines connecting $B$ and $C$ to $D$, and then extend the line $BD$ to a point $E$ on $AC$.
In $\triangle ABE$, $AB + AE>BE$, by Sum of Two Sides of Triangle Greater than Third Side.
Then, $AB + AC = AB + AE + EC > BE + EC$ by Euclid's second comm... | Given a [[Definition:Triangle (Geometry)|triangle]] and a [[Definition:Point|point]] inside it, the sum of the lengths of the [[Definition:Line Segment|line segments]] from the [[Definition:Endpoint of Line|endpoints]] of one [[Definition:Side of Polygon|side]] of the triangle to the point is less than the sum of the o... | :[[File:Point Inside Triangle.png|250px]]
Given a triangle $ABC$ and a point $D$ inside it.
We can [[Axiom:Euclid's First Postulate|construct]] lines connecting $B$ and $C$ to $D$, and then [[Axiom:Euclid's Second Postulate|extend]] the line $BD$ to a point $E$ on $AC$.
In $\triangle ABE$, $AB + AE>BE$, by [[Sum of ... | Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides | https://proofwiki.org/wiki/Lines_Through_Endpoints_of_One_Side_of_Triangle_to_Point_Inside_Triangle_is_Less_than_Sum_of_Other_Sides | https://proofwiki.org/wiki/Lines_Through_Endpoints_of_One_Side_of_Triangle_to_Point_Inside_Triangle_is_Less_than_Sum_of_Other_Sides | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Line/Segment",
"Definition:Line/Endpoint",
"Definition:Polygon/Side"
] | [
"File:Point Inside Triangle.png",
"Axiom:Euclid's First Postulate",
"Axiom:Euclid's Second Postulate",
"Sum of Two Sides of Triangle Greater than Third Side",
"Axiom:Euclid's Common Notions"
] |
proofwiki-1903 | Construction of Triangle from Given Lengths | Given three straight lines such that the sum of the lengths of any two of the lines is greater than the length of the third line, it is possible to construct a triangle having the lengths of these lines as its side lengths.
{{:Euclid:Proposition/I/22}} | Since $F$ is the center of the circle with radius $FD$, it follows from {{EuclidDefLink|I|15|Circle}} that $DF = KF$, so $a = KF$ by Euclid's first common notion.
Since $G$ is the center of the circle with radius $GH$, it follows from {{EuclidDefLink|I|15|Circle}} that $GH = GK$, so $c = GK$ by Euclid's first common no... | Given three [[Definition:Straight Line|straight lines]] such that the sum of the lengths of any two of the lines is greater than the length of the third line, it is possible to construct a [[Definition:Triangle (Geometry)|triangle]] having the lengths of these lines as its side lengths.
{{:Euclid:Proposition/I/22}} | Since $F$ is the center of the circle with radius $FD$, it follows from {{EuclidDefLink|I|15|Circle}} that $DF = KF$, so $a = KF$ by [[Axiom:Euclid's Common Notions|Euclid's first common notion]].
Since $G$ is the center of the circle with radius $GH$, it follows from {{EuclidDefLink|I|15|Circle}} that $GH = GK$, so $... | Construction of Triangle from Given Lengths | https://proofwiki.org/wiki/Construction_of_Triangle_from_Given_Lengths | https://proofwiki.org/wiki/Construction_of_Triangle_from_Given_Lengths | [
"Triangles"
] | [
"Definition:Line/Straight Line",
"Definition:Triangle (Geometry)"
] | [
"Axiom:Euclid's Common Notions",
"Axiom:Euclid's Common Notions",
"Sum of Two Sides of Triangle Greater than Third Side",
"Definition:Conditional/Necessary Condition",
"Construction of Equilateral Triangle"
] |
proofwiki-1904 | Connected Open Subset of Euclidean Space is Path-Connected | Let $\R^n$ be a Euclidean $n$-space.
Let $U$ be a connected open subset of $\R^n$.
Then $U$ is path-connected. | Let $a \in U$.
Let $H \subseteq U$ be the subset of points in $U$ which can be joined to $a$ by a path in $U$.
Let $K = U \setminus H$.
Let $x \in H$.
Then:
:$\exists \epsilon > 0: \map {B_\epsilon} x \subseteq U$
where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.
Given any $y \in \map {B_\epsilon} x$, the... | Let $\R^n$ be a [[Definition:Euclidean Space|Euclidean $n$-space]].
Let $U$ be a [[Definition:Connected Set (Topology)|connected]] [[Definition:Open Set (Topology)|open subset]] of $\R^n$.
Then $U$ is [[Definition:Path-Connected Set|path-connected]]. | Let $a \in U$.
Let $H \subseteq U$ be the [[Definition:Subset|subset]] of points in $U$ which can be joined to $a$ by a [[Definition:Path (Topology)|path]] in $U$.
Let $K = U \setminus H$.
Let $x \in H$.
Then:
:$\exists \epsilon > 0: \map {B_\epsilon} x \subseteq U$
where $\map {B_\epsilon} x$ is the [[Definition:... | Connected Open Subset of Euclidean Space is Path-Connected | https://proofwiki.org/wiki/Connected_Open_Subset_of_Euclidean_Space_is_Path-Connected | https://proofwiki.org/wiki/Connected_Open_Subset_of_Euclidean_Space_is_Path-Connected | [
"Connected Sets (Topology)",
"Path-Connected Sets",
"Real Euclidean Spaces"
] | [
"Definition:Euclidean Space",
"Definition:Connected Set (Topology)",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set"
] | [
"Definition:Subset",
"Definition:Path (Topology)",
"Definition:Open Ball",
"Definition:Line/Straight Line",
"Definition:Path (Topology)",
"Definition:Path (Topology)",
"Joining Paths makes Another Path",
"Definition:Path (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
... |
proofwiki-1905 | Joining Paths makes Another Path | Let $T$ be a topological space.
Let $I \subseteq \R$ be the closed real interval $\closedint 0 1$.
Let $f, g: I \to T$ be paths in $T$ from $a$ to $b$ and from $b$ to $c$ respectively.
Let $h: I \to T$ be the mapping given by:
$\map h x = \begin{cases}
\map f {2x} & : x \in \closedint 0 {\dfrac 1 2} \\
\map g {2x - 1} ... | First we see that $h$ is well-defined, because on $\closedint 0 {\dfrac 1 2} \cap \closedint {\dfrac 1 2} 1 = \set {\dfrac 1 2}$ we have $\map f 1 = b = \map g 0$.
Now $\mathbin h {\restriction_{\closedint 0 {\frac 1 2} } } \mathop = f \circ k$ where $k: \closedint 0 {\dfrac 1 2} \to \closedint 0 1$ is given by $\map k... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $I \subseteq \R$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 1$.
Let $f, g: I \to T$ be [[Definition:Path (Topology)|paths]] in $T$ from $a$ to $b$ and from $b$ to $c$ respectively.
Let $h: I \to T$ be the [[Defini... | First we see that $h$ is [[Definition:Well-Defined Mapping|well-defined]], because on $\closedint 0 {\dfrac 1 2} \cap \closedint {\dfrac 1 2} 1 = \set {\dfrac 1 2}$ we have $\map f 1 = b = \map g 0$.
Now $\mathbin h {\restriction_{\closedint 0 {\frac 1 2} } } \mathop = f \circ k$ where $k: \closedint 0 {\dfrac 1 2} \t... | Joining Paths makes Another Path | https://proofwiki.org/wiki/Joining_Paths_makes_Another_Path | https://proofwiki.org/wiki/Joining_Paths_makes_Another_Path | [
"Paths (Topology)",
"Path-Connectedness"
] | [
"Definition:Topological Space",
"Definition:Real Interval/Closed",
"Definition:Path (Topology)",
"Definition:Mapping",
"Definition:Path (Topology)"
] | [
"Definition:Well-Defined/Mapping",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Pasting Lemma/Finite Union of Closed Sets",
"Definition:Continuous Mapping (Topology)/Everywhere"
] |
proofwiki-1906 | Hinge Theorem | If two triangles have two pairs of sides which are the same length, the triangle with the larger included angle also has the larger third side.
{{:Euclid:Proposition/I/24}} | :250px
Let $\triangle ABC$ and $DEF$ be two triangles in which $AB = DE$, $AC = DF$, and $\angle CAB > \angle FDE$.
Construct $\angle EDG$ on $DE$ at point $D$.
Place $G$ so that $DG = AC$.
Join $EG$ and $FG$.
Since $AB = DE$, $\angle BAC = \angle EDG$, and $AC = DG$, by Triangle Side-Angle-Side Congruence:
:$BC = GE$
... | If two [[Definition:Triangle (Geometry)|triangles]] have two pairs of [[Definition:Side of Polygon|sides]] which are the same length, the triangle with the larger included angle also has the larger third side.
{{:Euclid:Proposition/I/24}} | :[[File:Hinge Theorem.png|250px]]
Let $\triangle ABC$ and $DEF$ be two [[Definition:Triangle (Geometry)|triangles]] in which $AB = DE$, $AC = DF$, and $\angle CAB > \angle FDE$.
[[Construction of Equal Angle|Construct $\angle EDG$]] on $DE$ at [[Definition:Point|point]] $D$.
[[Construction of Equal Straight Lines fr... | Hinge Theorem | https://proofwiki.org/wiki/Hinge_Theorem | https://proofwiki.org/wiki/Hinge_Theorem | [
"Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side"
] | [
"File:Hinge Theorem.png",
"Definition:Triangle (Geometry)",
"Construction of Equal Angle",
"Definition:Point",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Triangle Side-Angle-Side Congruence",
"Axiom:Euclid's Common Notions",
"Isosceles Triangle has Two Eq... |
proofwiki-1907 | Equivalence of Definitions of Component | {{TFAE|def = Component (Topology)|view = component|context = Topology (Mathematical Branch)|contextview = topology}}
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
=== Definition 1: Equivalence Class ===
{{:Definition:Component (Topology)/Definition 1}}
=== Definition 2: Union of Connected Sets ===
... | Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$
Let $C = \bigcup \CC_x$ | {{TFAE|def = Component (Topology)|view = component|context = Topology (Mathematical Branch)|contextview = topology}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
=== [[Definition:Component (Topology)/Definition 1|Definition 1: Equivalence Class]] ===
{{:Definit... | Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$
Let $C = \bigcup \CC_x$ | Equivalence of Definitions of Component | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component | [
"Equivalence of Definitions of Component",
"Components (Topology)"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)/Definition 1",
"Definition:Component (Topology)/Definition 2",
"Definition:Component (Topology)/Definition 3"
] | [] |
proofwiki-1908 | Converse Hinge Theorem | If two triangles have two pairs of sides which are the same length, the triangle in which the third side is longer also has the larger angle contained by the first two sides.
{{:Euclid:Proposition/I/25}} | :300px
Let $\triangle ABC$ and $\triangle DEF$ be two triangles in which:
:$AB = DF$
:$AC = DE$
:$BC > EF$
{{AimForCont}} that $\angle BAC \not > \angle EDF$.
Then either:
:$\angle BAC = \angle EDF$
or:
:$\angle BAC < \angle EDF$
Let $\angle BAC = \angle EDF$.
Then by Triangle Side-Angle-Side Congruence:
:$BC = EF$
But... | If two [[Definition:Triangle (Geometry)|triangles]] have two pairs of [[Definition:Side of Polygon|sides]] which are the same [[Definition:Length (Linear Measure)|length]], the [[Definition:Triangle (Geometry)|triangle]] in which the third [[Definition:Side of Polygon|side]] is longer also has the larger [[Definition:A... | :[[File:Converse Hinge Theorem.png|300px]]
Let $\triangle ABC$ and $\triangle DEF$ be two [[Definition:Triangle (Geometry)|triangles]] in which:
:$AB = DF$
:$AC = DE$
:$BC > EF$
{{AimForCont}} that $\angle BAC \not > \angle EDF$.
Then either:
:$\angle BAC = \angle EDF$
or:
:$\angle BAC < \angle EDF$
Let $\angle BA... | Converse Hinge Theorem | https://proofwiki.org/wiki/Converse_Hinge_Theorem | https://proofwiki.org/wiki/Converse_Hinge_Theorem | [
"Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Angle/Containment",
"Definition:Polygon/Side"
] | [
"File:Converse Hinge Theorem.png",
"Definition:Triangle (Geometry)",
"Triangle Side-Angle-Side Congruence",
"Proof by Contradiction",
"Greater Angle of Triangle Subtended by Greater Side",
"Proof by Contradiction"
] |
proofwiki-1909 | Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed | Let $M$ be a metric space.
Let $C \subseteq M$ be a subspace of $M$ which is sequentially compact in $M$.
Then $C$ is sequentially compact in itself {{iff}} $C$ is closed in $M$. | Follows directly from Closure of Subset of Metric Space by Convergent Sequence.
{{qed}}
Category:Metric Subspaces
Category:Sequentially Compact Spaces
mqj2iylshn2871w02l98raps0bmr2uw | Let $M$ be a [[Definition:Metric Space|metric space]].
Let $C \subseteq M$ be a [[Definition:Metric Subspace|subspace]] of $M$ which is [[Definition:Sequentially Compact Space|sequentially compact]] in $M$.
Then $C$ is [[Definition:Sequentially Compact In Itself|sequentially compact in itself]] {{iff}} $C$ is [[Defi... | Follows directly from [[Closure of Subset of Metric Space by Convergent Sequence]].
{{qed}}
[[Category:Metric Subspaces]]
[[Category:Sequentially Compact Spaces]]
mqj2iylshn2871w02l98raps0bmr2uw | Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed | https://proofwiki.org/wiki/Sequentially_Compact_Metric_Subspace_is_Sequentially_Compact_in_Itself_iff_Closed | https://proofwiki.org/wiki/Sequentially_Compact_Metric_Subspace_is_Sequentially_Compact_in_Itself_iff_Closed | [
"Metric Subspaces",
"Sequentially Compact Spaces"
] | [
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Sequentially Compact Space",
"Definition:Sequentially Compact Space/In Itself",
"Definition:Closed Set/Topology"
] | [
"Closure of Subset of Metric Space by Convergent Sequence",
"Category:Metric Subspaces",
"Category:Sequentially Compact Spaces"
] |
proofwiki-1910 | Invariance of Extremal Length under Conformal Mappings | Let $X, Y$ be Riemann surfaces (usually, subsets of the complex plane).
Let $\phi: X \to Y$ be a conformal isomorphism between $X$ and $Y$.
Let $\Gamma$ be a family of rectifiable curves (or, more generally, of unions of rectifiable curves) in $X$.
Let $\Gamma'$ be the family of their images under $\phi$.
Then $\Gamma$... | Let $\rho'$ be a conformal metric on $Y$ in the sense of the definition of extremal length, given in local coordinates as:
:$\map {\rho'} z \size {\d z}$
Let $\rho$ be the metric on $X$ obtained as the pull-back of this metric under $\phi$.
That is, $\rho$ is given in local coordinates as:
:$\map {\rho'} {\map \phi w} ... | Let $X, Y$ be [[Definition:Riemann Surface|Riemann surfaces]] (usually, [[Definition:Subset|subsets]] of the [[Definition:Complex Plane|complex plane]]).
Let $\phi: X \to Y$ be a [[Definition:Conformal Isomorphism|conformal isomorphism]] between $X$ and $Y$.
Let $\Gamma$ be a [[Definition:Indexed Family|family]] of [... | Let $\rho'$ be a [[Definition:Conformal Metric|conformal metric]] on $Y$ in the sense of the definition of [[Definition:Extremal Length|extremal length]], given in local coordinates as:
:$\map {\rho'} z \size {\d z}$
Let $\rho$ be the [[Definition:Metric|metric]] on $X$ obtained as the [[Definition:Pull-Back|pull-back... | Invariance of Extremal Length under Conformal Mappings | https://proofwiki.org/wiki/Invariance_of_Extremal_Length_under_Conformal_Mappings | https://proofwiki.org/wiki/Invariance_of_Extremal_Length_under_Conformal_Mappings | [
"Geometric Function Theory"
] | [
"Definition:Riemann Surface",
"Definition:Subset",
"Definition:Complex Number/Complex Plane",
"Definition:Conformal Isomorphism",
"Definition:Indexing Set/Family",
"Definition:Rectifiable Curve",
"Definition:Set Union",
"Definition:Rectifiable Curve",
"Definition:Indexing Set/Family",
"Definition:... | [
"Definition:Conformal Metric",
"Definition:Extremal Length",
"Definition:Metric Space/Metric",
"Definition:Pull-Back",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Definition:Extremal Length"
] |
proofwiki-1911 | Real Number Line is Complete Metric Space | The real number line $\R$ with the usual (Euclidean) metric forms a complete metric space. | From Real Number Line is Metric Space, the distance function defined as $\map d {x, y} = \size {x - y}$ is a metric on $\R$.
It remains to be shown that the metric space $\struct {\R, d}$ is complete.
By definition, this is done by demonstrating that every Cauchy sequence of real numbers has a limit.
This is demonstrat... | The [[Definition:Real Number Line with Euclidean Metric|real number line $\R$ with the usual (Euclidean) metric]] forms a [[Definition:Complete Metric Space|complete metric space]]. | From [[Real Number Line is Metric Space]], the [[Definition:Distance Function|distance function]] defined as $\map d {x, y} = \size {x - y}$ is a [[Definition:Metric|metric]] on $\R$.
It remains to be shown that the [[Definition:Metric Space|metric space]] $\struct {\R, d}$ is [[Definition:Complete Metric Space|comple... | Real Number Line is Complete Metric Space | https://proofwiki.org/wiki/Real_Number_Line_is_Complete_Metric_Space | https://proofwiki.org/wiki/Real_Number_Line_is_Complete_Metric_Space | [
"Real Number Line with Euclidean Metric",
"Examples of Complete Metric Spaces"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Complete Metric Space"
] | [
"Real Number Line is Metric Space",
"Definition:Distance Function",
"Definition:Metric Space/Metric",
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence",
"Definition:Real Number",
"Definition:Limit of Sequence/Metric Space",
"Cauchy's Convergence Criterion"
] |
proofwiki-1912 | Series Law for Extremal Length | Let $X$ be a Riemann surface.
Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.
Let every $\gamma \in \Gamma$ contain a $\gamma_1 \in \Gamma_1$ and a $\gamma_2 \in \Gamma_2$ such that $\gamma_1 \cap \gamma_2 = \O$.
Then the e... | Let $\rho_1 = \map {\rho_1} z \size {\d z}$ and $\rho_2 = \map {\rho_2} z \size {\d z}$ be conformal metrics as in the definition of extremal length.
It can be assumed that these are normalized:
:$\map A {\rho_j} = \map L {\Gamma_j, \rho_j}$ for $j \in \set {1, 2}$.
We define another metric $\rho = \map \rho z \size {... | Let $X$ be a [[Definition:Riemann Surface|Riemann surface]].
Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be families of [[Definition:Rectifiable Curve|rectifiable curves]] (or, more generally, families of unions of rectifiable curves) on $X$.
Let every $\gamma \in \Gamma$ contain a $\gamma_1 \in \Gamma_1$ and a $\gamma_2... | Let $\rho_1 = \map {\rho_1} z \size {\d z}$ and $\rho_2 = \map {\rho_2} z \size {\d z}$ be conformal metrics as in the [[Definition:Extremal Length|definition of extremal length]].
It can be assumed that these are [[Definition:Extremal Length#Normalizations|normalized]]:
:$\map A {\rho_j} = \map L {\Gamma_j, \rho_j}$ ... | Series Law for Extremal Length | https://proofwiki.org/wiki/Series_Law_for_Extremal_Length | https://proofwiki.org/wiki/Series_Law_for_Extremal_Length | [
"Geometric Function Theory"
] | [
"Definition:Riemann Surface",
"Definition:Rectifiable Curve",
"Definition:Extremal Length"
] | [
"Definition:Extremal Length",
"Definition:Extremal Length",
"Series Law for Extremal Length/Rho is Well Defined"
] |
proofwiki-1913 | Extremal Length of Union | Let $X$ be a Riemann surface.
Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.
Then the extremal length of their union satisfies:
:$\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } \le \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 ... | Set $\Gamma := \Gamma_1\cup \Gamma_2$.
Let $\rho_1$ and $\rho_2$ be conformal metrics as in the definition of extremal length, normalized such that:
:$\map L {\Gamma_1, \rho_1} = \map L {\Gamma_2, \rho_2} = 1$
We define a new metric by:
:$\rho := \map \max {\rho_1, \rho_2}$
{{explain|Prove that $\rho$ is a metric}}
The... | Let $X$ be a [[Definition:Riemann Surface|Riemann surface]].
Let $\Gamma_1$ and $\Gamma_2$ be families of [[Definition:Rectifiable Curve|rectifiable curves]] (or, more generally, families of unions of rectifiable curves) on $X$.
Then the [[Definition:Extremal Length|extremal length]] of their union satisfies:
:$\df... | Set $\Gamma := \Gamma_1\cup \Gamma_2$.
Let $\rho_1$ and $\rho_2$ be conformal metrics as in the [[Definition:Extremal Length|definition of extremal length]], [[Definition:Extremal Length#Normalizations|normalized]] such that:
:$\map L {\Gamma_1, \rho_1} = \map L {\Gamma_2, \rho_2} = 1$
We define a new metric by:
:$\r... | Extremal Length of Union | https://proofwiki.org/wiki/Extremal_Length_of_Union | https://proofwiki.org/wiki/Extremal_Length_of_Union | [
"Geometric Function Theory"
] | [
"Definition:Riemann Surface",
"Definition:Rectifiable Curve",
"Definition:Extremal Length"
] | [
"Definition:Extremal Length",
"Definition:Extremal Length",
"Category:Geometric Function Theory"
] |
proofwiki-1914 | Parallel Law for Extremal Length | Let $X$ be a Riemann surface.
Let $\Gamma_1, \Gamma_2$ be families of rectifiable curves (or, more generally, families of disjoint unions of rectifiable curves) on $X$.
Let $\Gamma_1$ and $\Gamma_2$ be disjoint, in the sense that:
:there exist disjoint Borel subsets $A_1, A_2 \subseteq X$ such that:
::for any $\gamma_1... | The assumption means that every element of $\Gamma_1 \cup \Gamma_2$ contains some element of $\Gamma$.
Hence:
{{begin-eqn }}
{{eqn | l =\frac 1 {\lambda \left({\Gamma}\right)}
| o = \ge
| r = \frac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)}
| c = Comparison Principle for Extremal Length
}}
{{eq... | Let $X$ be a [[Definition:Riemann Surface|Riemann surface]].
Let $\Gamma_1, \Gamma_2$ be families of [[Definition:Rectifiable Curve|rectifiable curves]] (or, more generally, families of disjoint unions of rectifiable curves) on $X$.
Let $\Gamma_1$ and $\Gamma_2$ be disjoint, in the sense that:
:there exist disjoint ... | The assumption means that every element of $\Gamma_1 \cup \Gamma_2$ contains some element of $\Gamma$.
Hence:
{{begin-eqn }}
{{eqn | l =\frac 1 {\lambda \left({\Gamma}\right)}
| o = \ge
| r = \frac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)}
| c = [[Comparison Principle for Extremal Length]]
}... | Parallel Law for Extremal Length | https://proofwiki.org/wiki/Parallel_Law_for_Extremal_Length | https://proofwiki.org/wiki/Parallel_Law_for_Extremal_Length | [
"Geometric Function Theory"
] | [
"Definition:Riemann Surface",
"Definition:Rectifiable Curve",
"Definition:Extremal Length"
] | [
"Comparison Principle for Extremal Length",
"Extremal Length of Union"
] |
proofwiki-1915 | Comparison Principle for Extremal Length | Let $X$ be a Riemann surface.
Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.
Let every element of $\Gamma_1$ contain some element of $\Gamma_2$.
Then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by:
:$\lambda \left(... | We have:
{{begin-eqn}}
{{eqn | l = L \left({\Gamma_1, \rho}\right)
| r = \inf_{\gamma \mathop \in \Gamma_1} L \left({\gamma, \rho}\right)
| c = by definition
}}
{{eqn | o = \ge
| r = \inf_{\gamma \mathop \in \Gamma_2} L \left({\gamma, \rho}\right)
| c = since every curve of $\Gamma_1$ contains a... | Let $X$ be a [[Definition:Riemann Surface|Riemann surface]].
Let $\Gamma_1$ and $\Gamma_2$ be families of [[Definition:Rectifiable Curve|rectifiable curves]] (or, more generally, families of [[Definition:Set Union|unions]] of [[Definition:Rectifiable Curve|rectifiable curves]]) on $X$.
Let every [[Definition:Element|... | We have:
{{begin-eqn}}
{{eqn | l = L \left({\Gamma_1, \rho}\right)
| r = \inf_{\gamma \mathop \in \Gamma_1} L \left({\gamma, \rho}\right)
| c = by definition
}}
{{eqn | o = \ge
| r = \inf_{\gamma \mathop \in \Gamma_2} L \left({\gamma, \rho}\right)
| c = since every curve of $\Gamma_1$ contains ... | Comparison Principle for Extremal Length | https://proofwiki.org/wiki/Comparison_Principle_for_Extremal_Length | https://proofwiki.org/wiki/Comparison_Principle_for_Extremal_Length | [
"Geometric Function Theory"
] | [
"Definition:Riemann Surface",
"Definition:Rectifiable Curve",
"Definition:Set Union",
"Definition:Rectifiable Curve",
"Definition:Element",
"Definition:Element",
"Definition:Extremal Length",
"Definition:Extremal Length"
] | [] |
proofwiki-1916 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | Let $M = \struct {X, d}$ be a metric space.
By {{Metric-space-axiom|2}}, we have:
:$\forall x, y, z \in X: \map d {x, y} + \map d {y, z} \ge \map d {x, z}$
By subtracting $\map d {y, z}$ from both sides:
:$\map d {x, y} \ge \map d {x, z} - \map d {y, z}$
Now we consider 2 cases.
;Case $1$: Suppose $\map d {x, z} - \ma... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
By {{Metric-space-axiom|2}}, we have:
:$\forall x, y, z \in X: \map d {x, y} + \map d {y, z} \ge \map d {x, z}$
By subtracting $\map d {y, z}$ from both sides:
:$\map d {x, y} \ge \map d {x, z} - \map d {y, z}$
Now we consider 2 cases.
;Case... | Reverse Triangle Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Definition:Metric Space"
] |
proofwiki-1917 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | From the Reverse Triangle Inequality:
:$\cmod {x - y} \ge \cmod {\cmod x - \cmod y}$
By the definition of both absolute value and complex modulus:
:$\cmod {\cmod x - \cmod y} \ge 0$
As:
:$\cmod x - \cmod y = \pm \cmod {\cmod x - \cmod y}$
it follows that:
:$\cmod {\cmod x - \cmod y} \ge \cmod x - \cmod y$
Hence the res... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | From the [[Reverse Triangle Inequality/Real and Complex Fields/Proof 1|Reverse Triangle Inequality]]:
:$\cmod {x - y} \ge \cmod {\cmod x - \cmod y}$
By the definition of both [[Definition:Absolute Value|absolute value]] and [[Definition:Complex Modulus|complex modulus]]:
:$\cmod {\cmod x - \cmod y} \ge 0$
As:
:$\cmod... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_1 | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Reverse Triangle Inequality/Real and Complex Fields/Proof 1",
"Definition:Absolute Value",
"Definition:Complex Modulus"
] |
proofwiki-1918 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | By the Triangle Inequality:
:$\cmod {x + y} - \cmod y \le \cmod x$
Let $z = x + y$.
Then $x = z - y$ and so:
:$\cmod z - \cmod y \le \cmod {z - y}$
Renaming variables as appropriate gives:
:$\cmod {x - y} \ge \cmod x - \cmod y$
{{qed}} | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | By the [[Triangle Inequality]]:
:$\cmod {x + y} - \cmod y \le \cmod x$
Let $z = x + y$.
Then $x = z - y$ and so:
:$\cmod z - \cmod y \le \cmod {z - y}$
Renaming variables as appropriate gives:
:$\cmod {x - y} \ge \cmod x - \cmod y$
{{qed}} | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_2 | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Triangle Inequality"
] |
proofwiki-1919 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | Let $z_1$ and $z_2$ be represented by the points $A$ and $B$ respectively in the complex plane.
From Geometrical Interpretation of Complex Subtraction, we can construct the parallelogram $OACB$ where:
:$OA$ and $OB$ represent $z_1$ and $z_2$ respectively
:$BA$ represents $z_1 - z_2$.
:400px
But $OA$, $OB$ and $BA$ form... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A$ and $B$ respectively in the [[Definition:Complex Plane|complex plane]].
From [[Geometrical Interpretation of Complex Subtraction]], we can construct the [[Definition:Parallelogram|parallelogram]] $OACB$ where:
:$OA$ and $OB$ represent $z_1$ and ... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 3 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_3 | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Geometrical Interpretation of Complex Subtraction",
"Definition:Quadrilateral/Parallelogram",
"File:Complex-Reverse-Triangle-Inequality-Corollary.png",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Sum of Two Sides of Tria... |
proofwiki-1920 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | Let $X$ denote either $\R$ or $\C$ as appropriate.
From Real Number Line is Metric Space and Complex Plane is Metric Space the distance function $d: X \times X \to \R$ can be defined as:
:$\map d {x, y} = \size {x - y}$
From the Reverse Triangle Inequality as applied to metric spaces:
:$(1): \quad \forall x, y, z \in X... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | Let $X$ denote either $\R$ or $\C$ as appropriate.
From [[Real Number Line is Metric Space]] and [[Complex Plane is Metric Space]] the [[Definition:Distance Function|distance function]] $d: X \times X \to \R$ can be defined as:
:$\map d {x, y} = \size {x - y}$
From the [[Reverse Triangle Inequality]] as applied to [... | Reverse Triangle Inequality/Real and Complex Fields/Proof 1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Proof_1 | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Real Number Line is Metric Space",
"Complex Plane is Metric Space",
"Definition:Distance Function",
"Reverse Triangle Inequality",
"Definition:Metric Space"
] |
proofwiki-1921 | Reverse Triangle Inequality | Let $M = \struct {X, d}$ be a metric space.
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== Normed Division Ring ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== Normed Vector Space ===
{{:Reverse Triangle Inequality/Normed Vector Space}}
=== Real and Complex Nu... | From proof $2$ of corollary $1$ to this result, which is derived independently:
:$\size {x - y} \ge \size x - \size y$
There are two cases:
$(1): \quad \size x \ge \size y$
We have :
:$\size {\size x - \size y} = \size x - \size y$
and the proof is finished.
{{qed|lemma}}
$(2): \quad \size y \ge \size x$
We have:
:$\si... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Then:
:$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
=== [[Reverse Triangle Inequality/Normed Division Ring|Normed Division Ring]] ===
{{:Reverse Triangle Inequality/Normed Division Ring}}
=== [[Reverse Tria... | From [[Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2|proof $2$ of corollary $1$ to this result]], which is derived independently:
:$\size {x - y} \ge \size x - \size y$
There are two cases:
$(1): \quad \size x \ge \size y$
We have :
:$\size {\size x - \size y} = \size x - \size y$
and the ... | Reverse Triangle Inequality/Real and Complex Fields/Proof 2 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Proof_2 | [
"Triangle Inequality",
"Named Theorems",
"Inequalities"
] | [
"Definition:Metric Space",
"Reverse Triangle Inequality/Normed Division Ring",
"Reverse Triangle Inequality/Normed Vector Space",
"Reverse Triangle Inequality/Real and Complex Fields"
] | [
"Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2",
"Negative of Absolute Value"
] |
proofwiki-1922 | Inverse Completion of Integral Domain Exists | Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.
Then an inverse completion of $\struct {D, \circ}$ can be constructed. | From the definition of an integral domain:
:All elements of $D^* = D \setminus \set {0_D}$ are cancellable
:$\struct {D^*, \circ}$ is a commutative semigroup.
So by the Inverse Completion Theorem, there exists an inverse completion of $\struct {D, \circ}$.
From Construction of Inverse Completion, this is done as follow... | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Ring Zero|zero]] is $0_D$ and whose [[Definition:Unity of Ring|unity]] is $1_D$.
Then an [[Definition:Inverse Completion|inverse completion]] of $\struct {D, \circ}$ can be constructed. | From the [[Definition:Integral Domain|definition of an integral domain]]:
:All elements of $D^* = D \setminus \set {0_D}$ are [[Definition:Cancellable Element|cancellable]]
:$\struct {D^*, \circ}$ is a [[Definition:Commutative Semigroup|commutative semigroup]].
So by the [[Inverse Completion Theorem]], there exists an... | Inverse Completion of Integral Domain Exists | https://proofwiki.org/wiki/Inverse_Completion_of_Integral_Domain_Exists | https://proofwiki.org/wiki/Inverse_Completion_of_Integral_Domain_Exists | [
"Integral Domains",
"Inverse Completions"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Inverse Completion"
] | [
"Definition:Integral Domain",
"Definition:Cancellable Element",
"Definition:Commutative Semigroup",
"Inverse Completion Theorem",
"Definition:Inverse Completion",
"Construction of Inverse Completion",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Construction of Inverse Comple... |
proofwiki-1923 | Zero of Inverse Completion of Integral Domain | Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.
Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$ as defined in Inverse Completion of Integral Domain Exists.
Let $x \in K: x = \dfrac p q$ such that $p = 0_D$.
Then $x$ is equal to the zero of $K$.
That is, ''any'' element ... | Let us define $\eqclass {\tuple {a, b} } \ominus$ as in the Inverse Completion of Integral Domain Exists.
That is, $\eqclass {\tuple {a, b} } \ominus$ is an equivalence class of elements of $D \times D^*$ under the congruence relation $\ominus$.
$\ominus$ is the congruence relation defined on $D \times D^*$ by:
:$\tupl... | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Ring Zero|zero]] is $0_D$.
Let $\struct {K, \circ}$ be the [[Definition:Inverse Completion|inverse completion]] of $\struct {D, \circ}$ as defined in [[Inverse Completion of Integral Domain Exists]].
Let $x \in K: x =... | Let us define $\eqclass {\tuple {a, b} } \ominus$ as in the [[Inverse Completion of Integral Domain Exists]].
That is, $\eqclass {\tuple {a, b} } \ominus$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Element|elements]] of $D \times D^*$ under the [[Definition:Congruence Relation|congruence ... | Zero of Inverse Completion of Integral Domain | https://proofwiki.org/wiki/Zero_of_Inverse_Completion_of_Integral_Domain | https://proofwiki.org/wiki/Zero_of_Inverse_Completion_of_Integral_Domain | [
"Integral Domains",
"Inverse Completions"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Inverse Completion",
"Inverse Completion of Integral Domain Exists",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Ring Zero"
] | [
"Inverse Completion of Integral Domain Exists",
"Definition:Equivalence Class",
"Definition:Element",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Inverse Completion of Integral Domain Exists",
"Equality of Division Products",
"Definition:Element",
"Definition:Element",
"De... |
proofwiki-1924 | Inverse Completion Less Zero of Integral Domain is Closed | Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.
Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$.
Then $\struct {K^*, \circ}$ is closed, where $K^* = K \setminus \set {0_K}$. | Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$.
We define $\struct {K, \circ}$ of $\struct {D, \circ}$ by means of the technique used in Inverse Completion of Integral Domain Exists.
The structure of $\struct {K, \circ}$ is such that each element of $\struct {K, \circ}$ is of the form $x \ci... | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Ring Zero|zero]] is $0_D$ and whose [[Definition:Unity of Ring|unity]] is $1_D$.
Let $\struct {K, \circ}$ be the [[Definition:Inverse Completion|inverse completion]] of $\struct {D, \circ}$.
Then $\struct {K^*, \circ}... | Let $\struct {K, \circ}$ be the [[Definition:Inverse Completion|inverse completion]] of $\struct {D, \circ}$.
We define $\struct {K, \circ}$ of $\struct {D, \circ}$ by means of the technique used in [[Inverse Completion of Integral Domain Exists]].
The structure of $\struct {K, \circ}$ is such that each [[Definition:... | Inverse Completion Less Zero of Integral Domain is Closed | https://proofwiki.org/wiki/Inverse_Completion_Less_Zero_of_Integral_Domain_is_Closed | https://proofwiki.org/wiki/Inverse_Completion_Less_Zero_of_Integral_Domain_is_Closed | [
"Integral Domains",
"Inverse Completions"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Inverse Completion",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Inverse Completion",
"Inverse Completion of Integral Domain Exists",
"Definition:Element",
"Zero of Inverse Completion of Integral Domain",
"Definition:Element",
"Product of Division Products",
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Zero... |
proofwiki-1925 | Composition of Mappings is Associative | The composition of mappings is an associative binary operation:
:$\paren {f_3 \circ f_2} \circ f_1 = f_3 \circ \paren {f_2 \circ f_1}$
where $f_1, f_2, f_3$ are arbitrary mappings which fulfil the conditions for the relevant compositions to be defined. | :<nowiki>$\begin{xy}
<0em,12em>*+{E} = "TL",
<12em,12em>*+{G} = "TR",
<0em,0em>*+{F} = "BL",
<12em,0em>*+{H} = "BR",
"TL";"TR" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_2 \circ f_1},
"BL";"BR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{f_3 \circ f_2},
"TL";"BL" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{f_1},
"BL";"TR" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f... | The [[Definition:Composition of Mappings|composition of mappings]] is an [[Definition:Associative Operation|associative]] [[Definition:Binary Operation|binary operation]]:
:$\paren {f_3 \circ f_2} \circ f_1 = f_3 \circ \paren {f_2 \circ f_1}$
where $f_1, f_2, f_3$ are arbitrary [[Definition:Mapping|mappings]] which fu... | :<nowiki>$\begin{xy}
<0em,12em>*+{E} = "TL",
<12em,12em>*+{G} = "TR",
<0em,0em>*+{F} = "BL",
<12em,0em>*+{H} = "BR",
"TL";"TR" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_2 \circ f_1},
"BL";"BR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{f_3 \circ f_2},
"TL";"BL" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{f_1},
"BL";"TR" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{... | Composition of Mappings is Associative | https://proofwiki.org/wiki/Composition_of_Mappings_is_Associative | https://proofwiki.org/wiki/Composition_of_Mappings_is_Associative | [
"Composite Mappings",
"Examples of Associative Operations"
] | [
"Definition:Composition of Mappings",
"Definition:Associative Operation",
"Definition:Operation/Binary Operation",
"Definition:Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Mapping",
"Definition:Composition of Relations",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Mapping",
"Definition:Composition of Relations",
"Definition:Domain (Set Theory)/Relation",
"Domain of Composite Relation",
"Domain of Composite Relation",
"Domain of Composite Relati... |
proofwiki-1926 | Group is Abelian iff it has Cross Cancellation Property | Let $G$ be a group.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G$ is abelian}}
{{item|(2):|$G$ has the cross cancellation property}}
{{end-itemize}} | Let us suppress the operation of $G$ for brevity. | Let $G$ be a [[Definition:Group|group]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G$ is [[Definition:Abelian Group|abelian]]}}
{{item|(2):|$G$ has the [[Definition:Cross Cancellation Property|cross cancellation property]]}}
{{end-itemize}} | Let us suppress the operation of $G$ for brevity. | Group is Abelian iff it has Cross Cancellation Property | https://proofwiki.org/wiki/Group_is_Abelian_iff_it_has_Cross_Cancellation_Property | https://proofwiki.org/wiki/Group_is_Abelian_iff_it_has_Cross_Cancellation_Property | [
"Abelian Groups"
] | [
"Definition:Group",
"Definition:Abelian Group",
"Definition:Cross Cancellation Property"
] | [] |
proofwiki-1927 | Group is Abelian iff it has Middle Cancellation Property | Let $G$ be a group.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G$ is abelian}}
{{item|(2):|$G$ satisfies the middle cancellation property}}
{{end-itemize}} | Let us suppress the operation of $G$ for brevity. | Let $G$ be a [[Definition:Group|group]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G$ is [[Definition:Abelian Group|abelian]]}}
{{item|(2):|$G$ satisfies the [[Definition:Middle Cancellation Property|middle cancellation property]]}}
{{end-itemize}} | Let us suppress the operation of $G$ for brevity. | Group is Abelian iff it has Middle Cancellation Property | https://proofwiki.org/wiki/Group_is_Abelian_iff_it_has_Middle_Cancellation_Property | https://proofwiki.org/wiki/Group_is_Abelian_iff_it_has_Middle_Cancellation_Property | [
"Abelian Groups"
] | [
"Definition:Group",
"Definition:Abelian Group",
"Definition:Middle Cancellation Property"
] | [] |
proofwiki-1928 | Rational Numbers form Totally Ordered Field | The set of rational numbers $\Q$ forms a totally ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$. | Recall that by Integers form Ordered Integral Domain, $\struct {\Z, +, \times, \le}$ is an ordered integral domain
By Rational Numbers form Field, $\struct {\Q, +, \times}$ is a field.
In the formal definition of rational numbers, $\struct {\Q, +, \times}$ is the field of quotients of $\struct {\Z, +, \times, \le}$
By ... | The [[Definition:Rational Number|set of rational numbers]] $\Q$ forms a [[Definition:Totally Ordered Field|totally ordered field]] under [[Definition:Rational Addition|addition]] and [[Definition:Rational Multiplication|multiplication]]: $\struct {\Q, +, \times, \le}$. | Recall that by [[Integers form Ordered Integral Domain]], $\struct {\Z, +, \times, \le}$ is an [[Definition:Ordered Integral Domain|ordered integral domain]]
By [[Rational Numbers form Field]], $\struct {\Q, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
In the [[Definition:Rational Number/Formal Def... | Rational Numbers form Totally Ordered Field | https://proofwiki.org/wiki/Rational_Numbers_form_Totally_Ordered_Field | https://proofwiki.org/wiki/Rational_Numbers_form_Totally_Ordered_Field | [
"Examples of Fields",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Totally Ordered Field",
"Definition:Addition/Rational Numbers",
"Definition:Multiplication/Rational Numbers"
] | [
"Integers form Ordered Integral Domain",
"Definition:Ordered Integral Domain",
"Rational Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Rational Number/Formal Definition",
"Definition:Field of Quotients",
"Total Ordering on Field of Quotients is Unique",
"Definition:Total Orde... |
proofwiki-1929 | Rational Numbers are Densely Ordered | Let $a, b \in \Q$ such that $a < b$.
Then $\exists c \in \Q: a < c < b$.
That is, the set of rational numbers is densely ordered. | From the definition of rational numbers, we can express $a$ and $b$ as $a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$.
Thus from Mediant is Between:
:$\dfrac {p_1} {q_1} < \dfrac {p_1 + p_2} {q_1 + q_2} < \dfrac {p_2} {q_2}$
From Rational Numbers form Field:
:$\dfrac {p_1 + p_2} {q_1 + q_2} \in \Q$
Hence $c = \dfrac ... | Let $a, b \in \Q$ such that $a < b$.
Then $\exists c \in \Q: a < c < b$.
That is, the [[Definition:Rational Number|set of rational numbers]] is [[Definition:Densely Ordered|densely ordered]]. | From the [[Definition:Rational Number|definition of rational numbers]], we can express $a$ and $b$ as $a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$.
Thus from [[Mediant is Between]]:
:$\dfrac {p_1} {q_1} < \dfrac {p_1 + p_2} {q_1 + q_2} < \dfrac {p_2} {q_2}$
From [[Rational Numbers form Field]]:
:$\dfrac {p_1 + p_... | Rational Numbers are Densely Ordered | https://proofwiki.org/wiki/Rational_Numbers_are_Densely_Ordered | https://proofwiki.org/wiki/Rational_Numbers_are_Densely_Ordered | [
"Rational Numbers",
"Densely Ordered"
] | [
"Definition:Rational Number",
"Definition:Densely Ordered"
] | [
"Definition:Rational Number",
"Mediant is Between",
"Rational Numbers form Field",
"Definition:Element",
"Definition:Strictly Between"
] |
proofwiki-1930 | Real Addition is Closed | The set of real numbers $\R$ is closed under addition:
:$\forall x, y \in \R: x + y \in \R$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Real Addition|addition]]:
:$\forall x, y \in \R: x + y \in \R$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Addition is Closed | https://proofwiki.org/wiki/Real_Addition_is_Closed | https://proofwiki.org/wiki/Real_Addition_is_Closed | [
"Real Addition",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Real Numbers"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Addition/Real Numbers"
] |
proofwiki-1931 | Real Addition is Well-Defined | The operation of addition on the set of real numbers $\R$ is well-defined. | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers:
:$\eqclass {\sequence {x_n} } {} \equiv \eqclass {\sequence {y_n} } {} \iff \forall \epsilon > 0: \exists n \in \N: \forall i, j > n: \size {x_i - y_j} < \epsilon$
Let $... | The operation of [[Definition:Real Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Well-Defined Operation|well-defined]]. | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]]:
:$\eqclass {\sequence {x_n} } {} \equiv \eqcla... | Real Addition is Well-Defined | https://proofwiki.org/wiki/Real_Addition_is_Well-Defined | https://proofwiki.org/wiki/Real_Addition_is_Well-Defined | [
"Real Addition"
] | [
"Definition:Addition/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Well-Defined/Operation"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Addition/Real Numbers",
"Triangle Inequality",
"Category:Real Addition"
] |
proofwiki-1932 | Real Multiplication is Well-Defined | The operation of multiplication on the set of real numbers $\R$ is well-defined. | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let:
:$x = \eqclass {\sequence {x_n} } {}$
:$y = \eqclass {\sequence {y_n} } {}$
where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equ... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Well-Defined Operation|well-defined]]. | From the definition, the [[Definition:Real Number|real numbers]] are the [[Definition:Set|set]] of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let:
:$x = \eqclass {\sequ... | Real Multiplication is Well-Defined | https://proofwiki.org/wiki/Real_Multiplication_is_Well-Defined | https://proofwiki.org/wiki/Real_Multiplication_is_Well-Defined | [
"Real Multiplication"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Well-Defined/Operation"
] | [
"Definition:Real Number",
"Definition:Set",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers",
"Definition:Cauchy Sequence",
"Cauchy Sequence is Bounded/Metric Space",
"Definition:Bou... |
proofwiki-1933 | Real Multiplication is Closed | The operation of multiplication on the set of real numbers $\R$ is closed:
:$\forall x, y \in \R: x \times y \in \R$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in \R: x \times y \in \R$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Multiplication is Closed | https://proofwiki.org/wiki/Real_Multiplication_is_Closed | https://proofwiki.org/wiki/Real_Multiplication_is_Closed | [
"Real Multiplication",
"Algebraic Closure"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers"
] |
proofwiki-1934 | Real Addition is Associative | The operation of addition on the set of real numbers $\R$ is associative:
:$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\seq... | The operation of [[Definition:Real Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Addition is Associative | https://proofwiki.org/wiki/Real_Addition_is_Associative | https://proofwiki.org/wiki/Real_Addition_is_Associative | [
"Real Addition",
"Examples of Associative Operations",
"Real Addition",
"Examples of Associative Operations",
"Associative Law of Addition"
] | [
"Definition:Addition/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Associative Operation"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Addition/Real Numbers",
"Rational Addition is Associative"
] |
proofwiki-1935 | Real Multiplication is Associative | The operation of multiplication on the set of real numbers $\R$ is associative:
:$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\seq... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Multiplication is Associative | https://proofwiki.org/wiki/Real_Multiplication_is_Associative | https://proofwiki.org/wiki/Real_Multiplication_is_Associative | [
"Real Multiplication",
"Examples of Associative Operations",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Associative Operation"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers",
"Rational Multiplication is Associative"
] |
proofwiki-1936 | Real Addition is Commutative | The operation of addition on the set of real numbers $\R$ is commutative:
:$\forall x, y \in \R: x + y = y + x$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The operation of [[Definition:Real Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \R: x + y = y + x$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Addition is Commutative | https://proofwiki.org/wiki/Real_Addition_is_Commutative | https://proofwiki.org/wiki/Real_Addition_is_Commutative | [
"Real Addition",
"Examples of Commutative Operations",
"Commutative Law of Addition"
] | [
"Definition:Addition/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Commutative/Operation"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Rational Addition is Commutative"
] |
proofwiki-1937 | Real Multiplication is Commutative | The operation of multiplication on the set of real numbers $\R$ is commutative:
:$\forall x, y \in \R: x \times y = y \times x$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \R: x \times y = y \times x$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Multiplication is Commutative | https://proofwiki.org/wiki/Real_Multiplication_is_Commutative | https://proofwiki.org/wiki/Real_Multiplication_is_Commutative | [
"Real Multiplication",
"Examples of Commutative Operations",
"Commutative Law of Multiplication"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Commutative/Operation"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers",
"Rational Multiplication is Commutative"
] |
proofwiki-1938 | Equidecomposable Nested Sets | Let $A, B, C$ be sets such that $A$ and $C$ are equidecomposable and $A \subseteq B \subseteq C$.
Then $B$ and $C$ are equidecomposable. | Let $\set {X_k}_{k \mathop = 1}^n$ be a decomposition of $A$ and $C$, so that there are isometries $\set {\phi_k}_{k \mathop = 1}^n$ and $\set {\psi_k}_{k \mathop = 1}^n$ such that:
$\ds A = \bigcup_{k \mathop = 1}^n \map {\phi_k} {X_k}$
$\ds C = \bigcup_{k \mathop = 1}^n \map {\psi_k} {X_k}$
Let $Y_k = \map {\psi_k^{-... | Let $A, B, C$ be [[Definition:Set|sets]] such that $A$ and $C$ are [[Definition:Equidecomposable Sets|equidecomposable]] and $A \subseteq B \subseteq C$.
Then $B$ and $C$ are [[Definition:Equidecomposable Sets|equidecomposable]]. | Let $\set {X_k}_{k \mathop = 1}^n$ be a [[Definition:Decomposition (Topology)|decomposition]] of $A$ and $C$, so that there are [[Definition:Isometry (Metric Spaces)|isometries]] $\set {\phi_k}_{k \mathop = 1}^n$ and $\set {\psi_k}_{k \mathop = 1}^n$ such that:
$\ds A = \bigcup_{k \mathop = 1}^n \map {\phi_k} {X_k}$
... | Equidecomposable Nested Sets | https://proofwiki.org/wiki/Equidecomposable_Nested_Sets | https://proofwiki.org/wiki/Equidecomposable_Nested_Sets | [
"Equidecomposable Sets",
"Topology"
] | [
"Definition:Set",
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets"
] | [
"Definition:Decomposable Set",
"Definition:Isometry (Metric Spaces)",
"Category:Equidecomposable Sets",
"Category:Topology"
] |
proofwiki-1939 | Rational Numbers form Metric Space | Let $\Q$ be the set of all rational numbers.
Let $d: \Q \times \Q \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
Then $d$ is a metric on $\Q$ and so $\struct {\Q, d}$ is a metric space. | From the definition of absolute value:
:$\size {x_1 - x_2} = \sqrt {\paren {x_1 - x_2}^2}$
This is the same as the Euclidean metric.
This is shown in Euclidean Metric on Real Vector Space is Metric to be a metric.
From Rational Numbers form Vector Space, it follows that the set of all rational numbers is a $1$-dimensio... | Let $\Q$ be the [[Definition:Set|set]] of all [[Definition:Rational Number|rational numbers]].
Let $d: \Q \times \Q \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$.
Then $d$ is a [[Definition:Metric|metric]] on $\Q$ and so $... | From the definition of [[Definition:Absolute Value|absolute value]]:
:$\size {x_1 - x_2} = \sqrt {\paren {x_1 - x_2}^2}$
This is the same as the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
This is shown in [[Euclidean Metric on Real Vector Space is Metric]] to be a [[Definition:Metric|metri... | Rational Numbers form Metric Space | https://proofwiki.org/wiki/Rational_Numbers_form_Metric_Space | https://proofwiki.org/wiki/Rational_Numbers_form_Metric_Space | [
"Rational Numbers",
"Rational Number Space",
"Examples of Euclidean Spaces",
"Examples of Metric Spaces"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Absolute Value",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [
"Definition:Absolute Value",
"Definition:Euclidean Metric/Real Number Line",
"Euclidean Metric on Real Vector Space is Metric",
"Definition:Metric Space/Metric",
"Rational Numbers form Vector Space",
"Definition:Set",
"Definition:Rational Number",
"Definition:Dimension (Geometry)",
"Definition:Eucli... |
proofwiki-1940 | Rational Numbers form Vector Space | Let $\Q$ be the set of rational numbers.
Then the $\Q$-module $\Q^n$ is a vector space.
It follows directly, by setting $n = 1$, that the $\Q$-module $\Q$ itself can also be regarded as a vector space. | From the definition, a vector space is a unitary module whose scalar ring is a field.
From Rational Numbers form Field, we have that $\Q$ is a field.
So the $\Q$-module $\Q^n$ fits the description.
{{qed}}
Category:Linear Algebra
Category:Rational Numbers
n74hux90kj7e88gtvcb4nzycbi1nbre | Let $\Q$ be the set of [[Definition:Rational Number|rational numbers]].
Then the [[Definition:Module on Cartesian Product|$\Q$-module $\Q^n$]] is a [[Definition:Vector Space|vector space]].
It follows directly, by setting $n = 1$, that the [[Definition:Module on Cartesian Product|$\Q$-module $\Q$]] itself can also ... | From the definition, a [[Definition:Vector Space|vector space]] is a [[Definition:Unitary Module|unitary module]] whose [[Definition:Scalar Ring of Unitary Module|scalar ring]] is a [[Definition:Field (Abstract Algebra)|field]].
From [[Rational Numbers form Field]], we have that $\Q$ is a [[Definition:Field (Abstract ... | Rational Numbers form Vector Space | https://proofwiki.org/wiki/Rational_Numbers_form_Vector_Space | https://proofwiki.org/wiki/Rational_Numbers_form_Vector_Space | [
"Linear Algebra",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Module on Cartesian Product",
"Definition:Vector Space",
"Definition:Module on Cartesian Product",
"Definition:Vector Space"
] | [
"Definition:Vector Space",
"Definition:Unitary Module over Ring",
"Definition:Scalar Ring/Unitary Module",
"Definition:Field (Abstract Algebra)",
"Rational Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Module on Cartesian Product",
"Category:Linear Algebra",
"Category:Ratio... |
proofwiki-1941 | Five Color Theorem | A planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 5$. | The proof proceeds by the Principle of Mathematical Induction on the number of vertices.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$G_n$ can be assigned a proper vertex $k$-coloring such that $k \le 5$. | A [[Definition:Planar Graph|planar graph]] $G$ can be assigned a [[Definition:Proper Vertex Coloring|proper vertex $k$-coloring]] such that $k \le 5$. | The proof proceeds by the [[Principle of Mathematical Induction]] on the number of [[Definition:Vertex of Graph|vertices]].
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$G_n$ can be assigned a [[Definition:Proper Vertex Coloring|proper vertex $k$-coloring]] such that $k \le... | Five Color Theorem | https://proofwiki.org/wiki/Five_Color_Theorem | https://proofwiki.org/wiki/Five_Color_Theorem | [
"Named Theorems",
"Graph Theory",
"5"
] | [
"Definition:Planar Graph",
"Definition:Proper Coloring/Vertex Coloring"
] | [
"Principle of Mathematical Induction",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Proposition",
"Definition:Proper Coloring/Vertex Coloring",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Proper Coloring/Vertex Coloring",
"Definition:Proper Coloring/Vertex Coloring",
"Definition:Gra... |
proofwiki-1942 | Limit of (Cosine (X) - 1) over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | This proof works directly from the definition of the cosine function:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}
| c = {{Defof|Real Cosine Function}}
}}
{{eqn | r = \paren {-1}^0 \cdot \frac {x^{2 \cdot 0} } {\paren {2 \cdot 0}!} + \sum_... | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | This proof works directly from the definition of the [[Definition:Real Cosine Function|cosine function]]:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}
| c = {{Defof|Real Cosine Function}}
}}
{{eqn | r = \paren {-1}^0 \cdot \frac {x^{2 \c... | Limit of (Cosine (X) - 1) over X at Zero/Proof 1 | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero/Proof_1 | [
"Limit of (Cosine (X) - 1) over X at Zero",
"Cosine Function",
"Examples of Limits of Real Functions",
"Differential Calculus"
] | [] | [
"Definition:Cosine/Real Function",
"Power Series is Differentiable on Interval of Convergence",
"L'Hôpital's Rule",
"Absolute Value Function is Completely Multiplicative",
"Power Function is Strictly Increasing over Positive Reals/Natural Exponent",
"Sum of Infinite Geometric Sequence",
"Weierstrass M-T... |
proofwiki-1943 | Limit of (Cosine (X) - 1) over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | This proof assumes the truth of the Derivative of Cosine Function:
From Cosine of Zero is One:
:$\cos 0 = 1$
From Derivative of Cosine Function:
:$\map {D_x} {\cos x} = -\sin x$
and by Derivative of Constant:
:$\map {D_x} {-1} = 0$
So by Sum Rule for Derivatives:
:$\map {D_x} {\cos x - 1} = -\sin x$
By Sine of Zero is ... | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | This proof assumes the truth of the [[Derivative of Cosine Function]]:
From [[Cosine of Zero is One]]:
:$\cos 0 = 1$
From [[Derivative of Cosine Function]]:
:$\map {D_x} {\cos x} = -\sin x$
and by [[Derivative of Constant]]:
:$\map {D_x} {-1} = 0$
So by [[Sum Rule for Derivatives]]:
:$\map {D_x} {\cos x - 1} = -\si... | Limit of (Cosine (X) - 1) over X at Zero/Proof 2 | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero/Proof_2 | [
"Limit of (Cosine (X) - 1) over X at Zero",
"Cosine Function",
"Examples of Limits of Real Functions",
"Differential Calculus"
] | [] | [
"Derivative of Cosine Function",
"Cosine of Zero is One",
"Derivative of Cosine Function",
"Derivative of Constant",
"Sum Rule for Derivatives",
"Sine of Zero is Zero",
"Derivative of Identity Function",
"L'Hôpital's Rule"
] |
proofwiki-1944 | Limit of (Cosine (X) - 1) over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\cos x - 1} x
| r = \lim_{x \mathop \to 0} \frac {\paren {\cos x - 1} \paren {\cos x + 1} } {x \paren {\cos x + 1} }
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {\cos^2 x - 1} {x \paren {\cos x + 1} }
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {-\sin^2 x} {... | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\cos x - 1} x
| r = \lim_{x \mathop \to 0} \frac {\paren {\cos x - 1} \paren {\cos x + 1} } {x \paren {\cos x + 1} }
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {\cos^2 x - 1} {x \paren {\cos x + 1} }
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {-\sin^2 x} {... | Limit of (Cosine (X) - 1) over X at Zero/Proof 3 | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero/Proof_3 | [
"Limit of (Cosine (X) - 1) over X at Zero",
"Cosine Function",
"Examples of Limits of Real Functions",
"Differential Calculus"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Limit of Sinc Function at Zero",
"Combination Theorem for Limits of Functions/Real/Quotient Rule"
] |
proofwiki-1945 | Limit of (Cosine (X) - 1) over X at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | {{begin-eqn}}
{{eqn | l = \frac {\cos x - 1} x
| r = \frac {\cos x - \cos 0} x
| c = Cosine of Zero is One
}}
{{eqn | o = \to
| r = \valueat {\dfrac \d {\d x} \cos x} {x \mathop = 0}
| c = as $x \to 0$, from {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \bigvalueat {\sin x} {x \... | :$\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ | {{begin-eqn}}
{{eqn | l = \frac {\cos x - 1} x
| r = \frac {\cos x - \cos 0} x
| c = [[Cosine of Zero is One]]
}}
{{eqn | o = \to
| r = \valueat {\dfrac \d {\d x} \cos x} {x \mathop = 0}
| c = as $x \to 0$, from {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \bigvalueat {\sin x} ... | Limit of (Cosine (X) - 1) over X at Zero/Proof 4 | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero | https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero/Proof_4 | [
"Limit of (Cosine (X) - 1) over X at Zero",
"Cosine Function",
"Examples of Limits of Real Functions",
"Differential Calculus"
] | [] | [
"Cosine of Zero is One",
"Derivative of Cosine Function",
"Sine of Zero is Zero"
] |
proofwiki-1946 | Real Multiplication Distributes over Addition | The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:
:$\forall x, y, z \in \R:$
::$x \times \paren {y + z} = x \times y + x \times z$
::$\paren {y + z} \times x = y \times x + z \times x$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\seq... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Distributive Operation|distributive]] over the operation of [[Definition:Real Addition|addition]]:
:$\forall x, y, z \in \R:$
::$x \times \paren {y + z} = x \t... | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Multiplication Distributes over Addition/Algebraic Proof | https://proofwiki.org/wiki/Real_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Real_Multiplication_Distributes_over_Addition/Algebraic_Proof | [
"Real Multiplication",
"Real Addition",
"Examples of Distributive Operations",
"Real Multiplication Distributes over Addition"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Distributive Operation",
"Definition:Addition/Real Numbers"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers",
"Definition:Addition/Real Numbers",
"Rational Multiplication Distributes over Addition",
"Real Addition is C... |
proofwiki-1947 | Real Multiplication Distributes over Addition | The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:
:$\forall x, y, z \in \R:$
::$x \times \paren {y + z} = x \times y + x \times z$
::$\paren {y + z} \times x = y \times x + z \times x$ | {{:Euclid:Proposition/II/1}}
:400px
Let $A$ and $BC$ be two straight lines.
Let $BC$ be cut at random at points $D$ and $E$.
Then the rectangle contained by $A$ and $BC$ is equal to the sum of the rectangles contained by $A$ and $BD$, by $A$ and $DE$, and by $A$ and $EC$, as follows:
Construct $BF$ perpendicular to $BC... | The operation of [[Definition:Real Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is [[Definition:Distributive Operation|distributive]] over the operation of [[Definition:Real Addition|addition]]:
:$\forall x, y, z \in \R:$
::$x \times \paren {y + z} = x \t... | {{:Euclid:Proposition/II/1}}
:[[File:Euclid-II-1.png|400px]]
Let $A$ and $BC$ be two [[Definition:Straight Line|straight lines]].
Let $BC$ be cut at random at points $D$ and $E$.
Then the [[Definition:Containment of Rectangle|rectangle contained]] by $A$ and $BC$ is equal to the sum of the [[Definition:Containment ... | Real Multiplication Distributes over Addition/Geometric Proof | https://proofwiki.org/wiki/Real_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Real_Multiplication_Distributes_over_Addition/Geometric_Proof | [
"Real Multiplication",
"Real Addition",
"Examples of Distributive Operations",
"Real Multiplication Distributes over Addition"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Distributive Operation",
"Definition:Addition/Real Numbers"
] | [
"File:Euclid-II-1.png",
"Definition:Line/Straight Line",
"Definition:Quadrilateral/Rectangle/Containment",
"Definition:Quadrilateral/Rectangle/Containment",
"Construction of Perpendicular Line",
"Construction of Equal Straight Lines from Unequal",
"Construction of Parallel Line",
"Construction of Para... |
proofwiki-1948 | Cardano's Formula | Let $P$ be the cubic equation:
:$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$
Then $P$ has solutions:
{{begin-eqn}}
{{eqn | l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | l = x_3
| r = -\dfrac {S + T} 2 -... | First the cubic is depressed, by using the Tschirnhaus Transformation:
:$x \to x + \dfrac b {3 a}$:
{{begin-eqn}}
{{eqn | l = a x^3 + b x^2 + c x + d
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x + \frac b {3 a} }^3 - 3 \frac b {3 a} x^2 - 3 \frac {b^2} {9 a^2} x - \frac {b^3} {27 a^3} + \fra... | Let $P$ be the [[Definition:Cubic Equation|cubic equation]]:
:$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$
Then $P$ has solutions:
{{begin-eqn}}
{{eqn | l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | l = x_3
... | First the cubic is [[Definition:Depressed Polynomial|depressed]], by using the [[Definition:Tschirnhaus Transformation|Tschirnhaus Transformation]]:
:$x \to x + \dfrac b {3 a}$:
{{begin-eqn}}
{{eqn | l = a x^3 + b x^2 + c x + d
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x + \frac b {3 a} }^... | Cardano's Formula | https://proofwiki.org/wiki/Cardano's_Formula | https://proofwiki.org/wiki/Cardano's_Formula | [
"Cubic Equations",
"Cardano's Formula"
] | [
"Definition:Cubic Equation"
] | [
"Definition:Depressed Polynomial",
"Definition:Tschirnhaus Transformation",
"Definition:Depressed Polynomial",
"Definition:Cubic Equation/Resolvent",
"Definition:Cubic Equation/Resolvent",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation",
"Definition:Square Root",
"Roots of Complex ... |
proofwiki-1949 | Ferrari's Method | Let $P$ be the quartic equation:
:$a x^4 + b x^3 + c x^2 + d x + e = 0$
such that $a \ne 0$.
Then $P$ has solutions:
:$x = \dfrac {-p \pm \sqrt {p^2 - 8 q} } 4$
where:
{{begin-eqn}}
{{eqn | l = p
| r = \dfrac b a \pm \sqrt {\dfrac {b^2} {a^2} - \dfrac {4 c} a + 4 y_1}
}}
{{eqn | l = q
| r = y_1 \mp \sqrt { ... | First we render the quartic into monic form:
:$x^4 + \dfrac b a x^3 + \dfrac c a x^2 + \dfrac d a x + \dfrac e a = 0$
Completing the Square in $x^2$:
:$\paren {x^2 + \dfrac b {2 a} x}^2 + \paren {\dfrac c a - \dfrac {b^2} {4 a^2} } x^2 + \dfrac d a x + \dfrac e a = 0$
Then we introduce a new variable $y$:
:$\paren {x^2... | Let $P$ be the [[Definition:Quartic Equation|quartic equation]]:
:$a x^4 + b x^3 + c x^2 + d x + e = 0$
such that $a \ne 0$.
Then $P$ has solutions:
:$x = \dfrac {-p \pm \sqrt {p^2 - 8 q} } 4$
where:
{{begin-eqn}}
{{eqn | l = p
| r = \dfrac b a \pm \sqrt {\dfrac {b^2} {a^2} - \dfrac {4 c} a + 4 y_1}
}}
{{eqn |... | First we render the [[Definition:Quartic Equation|quartic]] into [[Definition:Monic Polynomial|monic]] form:
:$x^4 + \dfrac b a x^3 + \dfrac c a x^2 + \dfrac d a x + \dfrac e a = 0$
[[Completing the Square]] in $x^2$:
:$\paren {x^2 + \dfrac b {2 a} x}^2 + \paren {\dfrac c a - \dfrac {b^2} {4 a^2} } x^2 + \dfrac d a x... | Ferrari's Method | https://proofwiki.org/wiki/Ferrari's_Method | https://proofwiki.org/wiki/Ferrari's_Method | [
"Ferrari's Method",
"Quartic Equations"
] | [
"Definition:Quartic Equation",
"Definition:Real Number",
"Definition:Cubic Equation",
"Ferrari's Method",
"Definition:Quartic Equation"
] | [
"Definition:Quartic Equation",
"Definition:Monic Polynomial",
"Completing the Square",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Cubic Equation",
"Cardano's Formula",
"Definition:Real Number",
"Definition:Quadratic Equation",
"Definition:Discriminant of Polynomial/Quadr... |
proofwiki-1950 | Prime Number Theorem | The prime-counting function $\map \pi n$, that is, the number of primes less than $n$, satisfies:
:$\ds \lim_{n \mathop \to \infty} \map \pi n \frac {\map \ln n} n = 1$
or equivalently:
:$\map \pi n \sim \dfrac n {\map \ln n}$
where $\sim$ denotes asymptotic equivalence. | {{questionable|The bounds obtained are too tight; the Landau notation calculation does not work}}
The proof presented here is a version of {{AuthorRef|Donald J. Newman}}'s proof. For ease of reading, the proof is broken into parts, with the goal of each part presented.
From the Von Mangoldt Equivalence, the '''Prime N... | The [[Definition:Prime-Counting Function|prime-counting function]] $\map \pi n$, that is, the number of [[Definition:Prime Number|primes]] less than $n$, satisfies:
:$\ds \lim_{n \mathop \to \infty} \map \pi n \frac {\map \ln n} n = 1$
or equivalently:
:$\map \pi n \sim \dfrac n {\map \ln n}$
where $\sim$ denotes [[D... | {{questionable|The bounds obtained are too tight; the Landau notation calculation does not work}}
The proof presented here is a version of {{AuthorRef|Donald J. Newman}}'s proof. For ease of reading, the proof is broken into parts, with the goal of each part presented.
From the [[Von Mangoldt Equivalence]], the '''[[... | Prime Number Theorem | https://proofwiki.org/wiki/Prime_Number_Theorem | https://proofwiki.org/wiki/Prime_Number_Theorem | [
"Prime Number Theorem",
"Prime Numbers",
"Named Theorems",
"Analytic Number Theory"
] | [
"Definition:Prime-Counting Function",
"Definition:Prime Number",
"Definition:Asymptotic Equality"
] | [
"Von Mangoldt Equivalence",
"Prime Number Theorem",
"Definition:Logical Equivalence",
"Definition:Von Mangoldt Function",
"Definition:Von Mangoldt Function",
"Zeta Equivalence to Prime Number Theorem",
"Prime Number Theorem",
"Definition:Logical Equivalence",
"Prime Number Theorem",
"Reciprocal of... |
proofwiki-1951 | Pointwise Inverse in Induced Structure | Let $\struct {G, \oplus}$ be a group whose identity is $e_G$.
Let $S$ be a set.
Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$.
Let $f \in G^S$.
Let $f^* \in G^S$ be the pointwise inverse of $f$:
:$\forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$
Then $f^*$ is the inverse of $f$ for th... | Let $f \in G^S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {f \oplus f^*} } x
| r = \map f x \oplus \map {f^*} x
| c =
}}
{{eqn | r = \map f x \oplus \paren {\map f x}^{-1}
| c =
}}
{{eqn | r = e_G
| c =
}}
{{end-eqn}}
Similarly for $\map {\paren {f^* \oplus f} } x$.
{{qed}} | Let $\struct {G, \oplus}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e_G$.
Let $S$ be a [[Definition:Set|set]].
Let $\struct {G^S, \oplus}$ be the structure on $G^S$ [[Definition:Induced Structure|induced]] by $\oplus$.
Let $f \in G^S$.
Let $f^* \in G^S$ be the [[Definition:... | Let $f \in G^S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {f \oplus f^*} } x
| r = \map f x \oplus \map {f^*} x
| c =
}}
{{eqn | r = \map f x \oplus \paren {\map f x}^{-1}
| c =
}}
{{eqn | r = e_G
| c =
}}
{{end-eqn}}
Similarly for $\map {\paren {f^* \oplus f} } x$.
{{qed}} | Pointwise Inverse in Induced Structure | https://proofwiki.org/wiki/Pointwise_Inverse_in_Induced_Structure | https://proofwiki.org/wiki/Pointwise_Inverse_in_Induced_Structure | [
"Inverse Elements",
"Pointwise Operations"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Pointwise Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Pointwise Operation"
] | [] |
proofwiki-1952 | Quaternion Group is Hamiltonian | The quaternion group $Q$ is Hamiltonian. | For clarity the Cayley table of $Q$ is presented below:
{{:Quaternion Group/Cayley Table}}
By definition $Q$ is Hamiltonian {{iff}}:
:$Q$ is non-abelian
and:
:every subgroup of $Q$ is normal.
$Q$ is non-abelian as demonstrated by the counter-example:
:$a b \ne b a$
From Subgroups of Quaternion Group:
{{:Subgroups of Qu... | The [[Definition:Quaternion Group|quaternion group]] $Q$ is [[Definition:Hamiltonian Group|Hamiltonian]]. | For clarity the [[Quaternion Group/Cayley Table|Cayley table of $Q$]] is presented below:
{{:Quaternion Group/Cayley Table}}
By definition $Q$ is [[Definition:Hamiltonian Group|Hamiltonian]] {{iff}}:
:$Q$ is [[Definition:Abelian Group|non-abelian]]
and:
:every [[Definition:Subgroup|subgroup]] of $Q$ is [[Definition:No... | Quaternion Group is Hamiltonian | https://proofwiki.org/wiki/Quaternion_Group_is_Hamiltonian | https://proofwiki.org/wiki/Quaternion_Group_is_Hamiltonian | [
"Quaternion Group",
"Hamiltonian Groups"
] | [
"Definition:Dicyclic Group/Quaternion Group",
"Definition:Hamiltonian Group"
] | [
"Quaternion Group/Cayley Table",
"Definition:Hamiltonian Group",
"Definition:Abelian Group",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Abelian Group",
"Quaternion Group/Subgroups",
"Trivial Subgroup and Group Itself are Normal",
"Definition:Normal Subgroup",
"Center of Quate... |
proofwiki-1953 | Finite Subset of Metric Space has no Limit Points | Let $M = \struct {A, d}$ be a metric space.
Let $X \subseteq A$ such that $X$ is finite.
Then $X$ has no limit points. | Let $x \in X$.
From Point in Finite Metric Space is Isolated, $x$ is an isolated point.
The result follows by definition of isolated point:
:$x$ is an isolated point {{iff}} $x$ is not a limit point.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $X \subseteq A$ such that $X$ is [[Definition:Finite|finite]].
Then $X$ has no [[Definition:Limit Point (Metric Space)|limit points]]. | Let $x \in X$.
From [[Point in Finite Metric Space is Isolated]], $x$ is an [[Definition:Isolated Point of Subset|isolated point]].
The result follows by definition of [[Definition:Isolated Point of Subset/Definition 2|isolated point]]:
:$x$ is an [[Definition:Isolated Point of Subset|isolated point]] {{iff}} $x$ is ... | Finite Subset of Metric Space has no Limit Points | https://proofwiki.org/wiki/Finite_Subset_of_Metric_Space_has_no_Limit_Points | https://proofwiki.org/wiki/Finite_Subset_of_Metric_Space_has_no_Limit_Points | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Finite",
"Definition:Limit Point/Metric Space"
] | [
"Point in Finite Metric Space is Isolated",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point of Subset/Definition 2",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Limit Point/Metric Space"
] |
proofwiki-1954 | Point in Finite Metric Space is Isolated | Let $M = \struct {A, d}$ be a metric space.
Let $X \subseteq A$ such that $X$ is finite.
Let $x \in X$.
Then $x$ is isolated in $X$. | As $X$ is finite, its elements can be placed in one-to-one correspondence with the elements of $\N^*_n$ for some $n \in \N$.
So let $X = \left\{{x_1, x_2, \ldots, x_n}\right\}$.
Now let $\epsilon := \min \left\{{\forall i, j \in \N^*_n: i \ne j: d \left({x_i, x_j}\right)}\right\}$.
That is, $\epsilon$ is the minimum di... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $X \subseteq A$ such that $X$ is [[Definition:Finite Set|finite]].
Let $x \in X$.
Then $x$ is [[Definition:Isolated Point (Metric Space)|isolated in $X$]]. | As $X$ is [[Definition:Finite Set|finite]], its [[Definition:Element|elements]] can be placed in [[Definition:Bijection|one-to-one correspondence]] with the [[Definition:Element|elements]] of $\N^*_n$ for some $n \in \N$.
So let $X = \left\{{x_1, x_2, \ldots, x_n}\right\}$.
Now let $\epsilon := \min \left\{{\forall i... | Point in Finite Metric Space is Isolated/Proof 1 | https://proofwiki.org/wiki/Point_in_Finite_Metric_Space_is_Isolated | https://proofwiki.org/wiki/Point_in_Finite_Metric_Space_is_Isolated/Proof_1 | [
"Isolated Points",
"Finite Metric Spaces",
"Point in Finite Metric Space is Isolated"
] | [
"Definition:Metric Space",
"Definition:Finite Set",
"Definition:Isolated Point (Metric Space)"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Bijection",
"Definition:Element",
"Definition:Distance Function",
"Definition:Element",
"Definition:Metric Space/Metric",
"Definition:Open Ball",
"Definition:Isolated Point (Metric Space)"
] |
proofwiki-1955 | Point in Finite Metric Space is Isolated | Let $M = \struct {A, d}$ be a metric space.
Let $X \subseteq A$ such that $X$ is finite.
Let $x \in X$.
Then $x$ is isolated in $X$. | A Metric Space is Hausdorff.
Therefore Point in Finite Hausdorff Space is Isolated can be applied.
Hence the result.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $X \subseteq A$ such that $X$ is [[Definition:Finite Set|finite]].
Let $x \in X$.
Then $x$ is [[Definition:Isolated Point (Metric Space)|isolated in $X$]]. | A [[Metric Space is Hausdorff]].
Therefore [[Point in Finite Hausdorff Space is Isolated]] can be applied.
Hence the result.
{{qed}} | Point in Finite Metric Space is Isolated/Proof 2 | https://proofwiki.org/wiki/Point_in_Finite_Metric_Space_is_Isolated | https://proofwiki.org/wiki/Point_in_Finite_Metric_Space_is_Isolated/Proof_2 | [
"Isolated Points",
"Finite Metric Spaces",
"Point in Finite Metric Space is Isolated"
] | [
"Definition:Metric Space",
"Definition:Finite Set",
"Definition:Isolated Point (Metric Space)"
] | [
"Metric Space is T2",
"Point in Finite T2 Space is Isolated"
] |
proofwiki-1956 | Point in Finite T2 Space is Isolated | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $X \subseteq S$ such that $X$ is finite.
Let $x \in X$.
Then $x$ is isolated in $X$. | As $X$ is finite, its elements can be placed in one-to-one correspondence with the elements of $\set {1, 2, \ldots, n}$ for some $n \in \N$.
So let $X = \set {x_1, x_2, \ldots, x_n}$.
From the definition of $T_2$ space:
: $\forall x_i, x_j \in X: x_i \ne x_j: \exists U, V \in \tau: x_i \in U, x_j \in V: U \cap V = \O$
... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $X \subseteq S$ such that $X$ is [[Definition:Finite Set|finite]].
Let $x \in X$.
Then $x$ is [[Definition:Isolated Point (Topology)|isolated in $X$]]. | As $X$ is [[Definition:Finite Set|finite]], its [[Definition:Element|elements]] can be placed in [[Definition:Bijection|one-to-one correspondence]] with the elements of $\set {1, 2, \ldots, n}$ for some $n \in \N$.
So let $X = \set {x_1, x_2, \ldots, x_n}$.
From the definition of [[Definition:T2 Space|$T_2$ space]]:
... | Point in Finite T2 Space is Isolated/Proof 1 | https://proofwiki.org/wiki/Point_in_Finite_T2_Space_is_Isolated | https://proofwiki.org/wiki/Point_in_Finite_T2_Space_is_Isolated/Proof_1 | [
"Point in Finite T2 Space is Isolated",
"Finite Topological Spaces",
"Isolated Points",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Finite Set",
"Definition:Isolated Point (Topology)"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Bijection",
"Definition:T2 Space",
"Definition:Topology",
"Definition:Set Intersection",
"Definition:Finite Set",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)"
] |
proofwiki-1957 | Point in Finite T2 Space is Isolated | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $X \subseteq S$ such that $X$ is finite.
Let $x \in X$.
Then $x$ is isolated in $X$. | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $X \subseteq T$ be finite.
From Separation Properties Preserved in Subspace, it follows that $\struct {X, \tau_X}$ is also a $T_2$ space.
From $T_2$ Space is $T_1$ Space it follows that $\struct {X, \tau_X}$ is a $T_1$ space.
From Finite $T_1$ Space is Discr... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $X \subseteq S$ such that $X$ is [[Definition:Finite Set|finite]].
Let $x \in X$.
Then $x$ is [[Definition:Isolated Point (Topology)|isolated in $X$]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $X \subseteq T$ be [[Definition:Finite Set|finite]].
From [[Separation Properties Preserved in Subspace]], it follows that $\struct {X, \tau_X}$ is also a [[Definition:T2 Space|$T_2$ space]].
From [[T2 Space is T1 Space|$T_2$ Space... | Point in Finite T2 Space is Isolated/Proof 2 | https://proofwiki.org/wiki/Point_in_Finite_T2_Space_is_Isolated | https://proofwiki.org/wiki/Point_in_Finite_T2_Space_is_Isolated/Proof_2 | [
"Point in Finite T2 Space is Isolated",
"Finite Topological Spaces",
"Isolated Points",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Finite Set",
"Definition:Isolated Point (Topology)"
] | [
"Definition:T2 Space",
"Definition:Finite Set",
"Separation Properties Preserved in Subspace",
"Definition:T2 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Finite T1 Space is Discrete",
"Definition:Discrete Topology",
"Topological Space is Discrete iff All Points are Isolated"
] |
proofwiki-1958 | Subsequence of Sequence in Metric Space with Limit | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Let $x$ be a limit point of $S = \set {x_n: n \in \N}$, the set of members of $\sequence {x_n}$.
Then $\sequence {x_n}$ has a subsequence which converges to $x$. | By Finite Subset of Metric Space has no Limit Points, $S$ is infinite (or it has no limit points).
We may assume that $x_n$ never equals $x$.
Otherwise we delete all instances of $x$ from $\sequence {x_n}$ and create a new sequence $x_m$ which is a subsequence of $x_n$ which ''does'' never equal $x$.
Then $S \setminus ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $M$]].
Let $x$ be a [[Definition:Limit Point (Metric Space)|limit point]] of $S = \set {x_n: n \in \N}$, the set of members of $\sequence {x_n}$.
Then $\sequence {x_n}$ has a [[Defini... | By [[Finite Subset of Metric Space has no Limit Points]], $S$ is [[Definition:Infinite|infinite]] (or it has no limit points).
We may assume that $x_n$ never equals $x$.
Otherwise we delete all instances of $x$ from $\sequence {x_n}$ and create a new sequence $x_m$ which is a subsequence of $x_n$ which ''does'' neve... | Subsequence of Sequence in Metric Space with Limit | https://proofwiki.org/wiki/Subsequence_of_Sequence_in_Metric_Space_with_Limit | https://proofwiki.org/wiki/Subsequence_of_Sequence_in_Metric_Space_with_Limit | [
"Metric Spaces",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Limit Point/Metric Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Metric Space"
] | [
"Finite Subset of Metric Space has no Limit Points",
"Definition:Infinite",
"Definition:Integer",
"Definition:Open Ball",
"Definition:Subsequence"
] |
proofwiki-1959 | Primes of form Power Less One | Let $m, n \in \N_{>0}$ be natural numbers.
Let $m^n - 1$ be prime.
Then $m = 2$ and $n$ is prime. | First we note that by Integer Less One divides Power Less One:
:$\paren {m - 1} \divides \paren {m^n - 1}$
where $\divides$ denotes divisibility.
Thus $m^n - 1$ is composite for all $m \in \Z: m > 2$.
Let $m = 2$, and consider $2^n - 1$.
Suppose $n$ is composite.
Then $n = r s$ where $r, s \in \Z_{> 1}$.
Then by {{Coro... | Let $m, n \in \N_{>0}$ be [[Definition:Natural Numbers|natural numbers]].
Let $m^n - 1$ be [[Definition:Prime Number|prime]].
Then $m = 2$ and $n$ is [[Definition:Prime Number|prime]]. | First we note that by [[Integer Less One divides Power Less One]]:
:$\paren {m - 1} \divides \paren {m^n - 1}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Thus $m^n - 1$ is [[Definition:Composite Number|composite]] for all $m \in \Z: m > 2$.
Let $m = 2$, and consider $2^n - 1$.
Suppose... | Primes of form Power Less One | https://proofwiki.org/wiki/Primes_of_form_Power_Less_One | https://proofwiki.org/wiki/Primes_of_form_Power_Less_One | [
"Mersenne Primes",
"Prime Numbers",
"Mersenne Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Prime Number"
] | [
"Integer Less One divides Power Less One",
"Definition:Divisor (Algebra)/Integer",
"Definition:Composite Number",
"Definition:Composite Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-1960 | Lebesgue's Number Lemma/Sequentially Compact Space | Let $M = \struct {A, d}$ be a metric space.
Let $M$ be sequentially compact.
Then there exists a Lebesgue number for every open cover of $M$. | {{AimForCont}} $\UU$ is an open cover of $M$ for which no Lebesgue number exists.
Then for any $n \in \N$, there exists some $x_n \in M$ such that $\map {B_{1 / n} } {x_n} \subseteq U$ is false for each $U \in \UU$. (Otherwise $1/n$ would be a Lebesgue number for $\UU$.)
As $M$ is sequentially compact, $\sequence {x_n}... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Sequentially Compact Space|sequentially compact]].
Then there exists a [[Definition:Lebesgue Number|Lebesgue number]] for every [[Definition:Open Cover|open cover]] of $M$. | {{AimForCont}} $\UU$ is an [[Definition:Open Cover|open cover]] of $M$ for which no [[Definition:Lebesgue Number|Lebesgue number]] exists.
Then for any $n \in \N$, there exists some $x_n \in M$ such that $\map {B_{1 / n} } {x_n} \subseteq U$ is false for each $U \in \UU$. (Otherwise $1/n$ would be a [[Definition:Lebes... | Lebesgue's Number Lemma/Sequentially Compact Space | https://proofwiki.org/wiki/Lebesgue's_Number_Lemma/Sequentially_Compact_Space | https://proofwiki.org/wiki/Lebesgue's_Number_Lemma/Sequentially_Compact_Space | [
"Lebesgue's Number Lemma",
"Sequentially Compact Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequentially Compact Space",
"Definition:Lebesgue Number",
"Definition:Open Cover"
] | [
"Definition:Open Cover",
"Definition:Lebesgue Number",
"Definition:Lebesgue Number",
"Definition:Sequentially Compact Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Open Cover",
"Definition:Open Set/Metric Space",
"Definition:Contradiction",
"Proof by ... |
proofwiki-1961 | First Subsequence Rule | Let $T = \struct {A, \tau}$ be a Hausdorff space.
Let $\sequence {x_n}$ be a sequence in $T$.
Suppose $\sequence {x_n}$ has two convergent subsequences with different limits.
Then $\sequence {x_n}$ is divergent. | As stated, let $T = \struct {A, \tau}$ be a Hausdorff space.
Let $\sequence {x_n}$ be a sequence in $T$.
Let $\sequence {y_n}$ and $\sequence {z_n}$ be convergent subsequences of $\sequence {x_n}$ with different limits.
{{AimForCont}} $\sequence {x_n}$ is convergent.
From Convergent Sequence in Hausdorff Space has Uniq... | Let $T = \struct {A, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $T$.
Suppose $\sequence {x_n}$ has two [[Definition:Convergent Sequence (Topology)|convergent]] [[Definition:Subsequence|subsequences]] with different [[Definition:Limit of S... | As stated, let $T = \struct {A, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $T$.
Let $\sequence {y_n}$ and $\sequence {z_n}$ be [[Definition:Convergent Sequence (Topology)|convergent]] [[Definition:Subsequence|subsequences]] of $\sequence... | First Subsequence Rule | https://proofwiki.org/wiki/First_Subsequence_Rule | https://proofwiki.org/wiki/First_Subsequence_Rule | [
"Hausdorff Spaces",
"Named Theorems"
] | [
"Definition:T2 Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Subsequence",
"Definition:Limit of Sequence/Topological Space",
"Definition:Divergent Sequence"
] | [
"Definition:T2 Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Subsequence",
"Definition:Limit of Sequence/Topological Space",
"Definition:Convergent Sequence/Topology",
"Convergent Sequence in T2 Space has Unique Limit",
"Definition:Limit of Sequence/Topological ... |
proofwiki-1962 | Second Subsequence Rule | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Suppose $\sequence {x_n}$ has a subsequence which is unbounded.
Then $\sequence {x_n}$ is divergent. | Follows by the Rule of Transposition from Convergent Sequence is Bounded.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $M$]].
Suppose $\sequence {x_n}$ has a [[Definition:Subsequence|subsequence]] which is [[Definition:Unbounded Sequence|unbounded]].
Then $\sequence {x_n}$ is [[Definition:Divergent Seq... | Follows by the [[Rule of Transposition]] from [[Convergent Sequence is Bounded]].
{{qed}} | Second Subsequence Rule | https://proofwiki.org/wiki/Second_Subsequence_Rule | https://proofwiki.org/wiki/Second_Subsequence_Rule | [
"Metric Spaces",
"Named Theorems"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Subsequence",
"Definition:Bounded Sequence/Unbounded",
"Definition:Divergent Sequence"
] | [
"Rule of Transposition",
"Convergent Sequence in Metric Space is Bounded"
] |
proofwiki-1963 | Open Set Less One Point is Open | Let $M = \struct {A, d}$ be a metric space.
Let $U \subseteq M$ be an open set of $M$.
Let $\alpha \in U$.
Then $U \setminus \set \alpha$ is open in $M$. | Let $x \in U \setminus \set \alpha$.
Let $\delta = \map d {x, \alpha}$.
Let $\map {B_\epsilon} x \subseteq U$ be an open $\epsilon$-ball of $x$ in $U$.
Let $\zeta = \min \set {\epsilon, \delta}$.
Then:
:$\map {B_\epsilon} x \subseteq U \setminus \set \alpha$
The result follows.
{{qed}}
Category:Open Set Less One Point ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $U \subseteq M$ be an [[Definition:Open Set (Metric Space)|open set]] of $M$.
Let $\alpha \in U$.
Then $U \setminus \set \alpha$ is [[Definition:Open Set (Metric Space)|open]] in $M$. | Let $x \in U \setminus \set \alpha$.
Let $\delta = \map d {x, \alpha}$.
Let $\map {B_\epsilon} x \subseteq U$ be an [[Definition:Open Ball|open $\epsilon$-ball]] of $x$ in $U$.
Let $\zeta = \min \set {\epsilon, \delta}$.
Then:
:$\map {B_\epsilon} x \subseteq U \setminus \set \alpha$
The result follows.
{{qed}}
[[... | Open Set Less One Point is Open | https://proofwiki.org/wiki/Open_Set_Less_One_Point_is_Open | https://proofwiki.org/wiki/Open_Set_Less_One_Point_is_Open | [
"Open Set Less One Point is Open",
"Open Sets (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Definition:Open Ball",
"Category:Open Set Less One Point is Open",
"Category:Open Sets (Metric Spaces)"
] |
proofwiki-1964 | Divisibility by 7 | An integer $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is divisible by $7$ {{iff}}:
:$\ds \sum_{i \mathop = 0}^{n - 1} \paren {3^i X_i}$ is divisible by $7$. | {{begin-eqn}}
{{eqn | l = X
| r = 10^0 X_0 + 10^1 X_1 + \cdots + 10^{n - 1} X_{n-1}
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} 10^i X_i
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} \paren {10^i - 3^i + 3^i} X_i
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} \paren {\paren {10^i - 3^i} X_i} + \sum_{i \mathop = 0}^{n ... | An [[Definition:Integer|integer]] $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is [[Definition:Divisor of Integer|divisible]] by $7$ {{iff}}:
:$\ds \sum_{i \mathop = 0}^{n - 1} \paren {3^i X_i}$ is [[Definition:Divisor of Integer|divisible]] by $7$. | {{begin-eqn}}
{{eqn | l = X
| r = 10^0 X_0 + 10^1 X_1 + \cdots + 10^{n - 1} X_{n-1}
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} 10^i X_i
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} \paren {10^i - 3^i + 3^i} X_i
}}
{{eqn | r = \sum_{i \mathop = 0}^{n - 1} \paren {\paren {10^i - 3^i} X_i} + \sum_{i \mathop = 0}^{n ... | Divisibility by 7 | https://proofwiki.org/wiki/Divisibility_by_7 | https://proofwiki.org/wiki/Divisibility_by_7 | [
"Divisibility by 7",
"Divisibility Tests",
"7"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Difference of Two Powers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-1965 | Finite Subspace of Dense-in-itself Metric Space is not Open | Let $M = \struct {A, d}$ be a metric space that is dense-in-itself.
Let $U$ be a finite subset of $A$.
Then $U$ is not an open set of $M$. | Let $U = \set {x_0, x_1, \ldots, x_n}$.
{{AimForCont}} $U$ is open.
Let:
:$\ds D = \min_{i \mathop \ne j} \map d {x_i, x_j}$
Let $x_j \in U$.
Consider the open ball $\map {B_{D/2} } {x_j}$ of $x_j$.
From Open Ball of Metric Space is Open Set, $\map {B_{D/2} } {x_j}$ is an open set.
Then from Finite Intersection of Open... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] that is [[Definition:Dense-in-itself|dense-in-itself]].
Let $U$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $A$.
Then $U$ is not an [[Definition:Open Set (Metric Space)|open set]] of $M$. | Let $U = \set {x_0, x_1, \ldots, x_n}$.
{{AimForCont}} $U$ is [[Definition:Open Set (Metric Space)|open]].
Let:
:$\ds D = \min_{i \mathop \ne j} \map d {x_i, x_j}$
Let $x_j \in U$.
Consider the [[Definition:Open Ball|open ball]] $\map {B_{D/2} } {x_j}$ of $x_j$.
From [[Open Ball of Metric Space is Open Set]], $\ma... | Finite Subspace of Dense-in-itself Metric Space is not Open | https://proofwiki.org/wiki/Finite_Subspace_of_Dense-in-itself_Metric_Space_is_not_Open | https://proofwiki.org/wiki/Finite_Subspace_of_Dense-in-itself_Metric_Space_is_not_Open | [
"Dense-in-itself",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Dense-in-itself",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Open Set/Metric Space"
] | [
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Open Ball is Open Set/Pseudometric Space",
"Definition:Open Set/Metric Space",
"Finite Intersection of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Definition:Singleton",
"Definition:Isol... |
proofwiki-1966 | Region Less One Point is Region | Let $M = \struct {A, d}$ be a dense-in-itself metric space.
Let $R \subseteq M$ be a region of $M$.
Let $x \in R$.
Then $R \setminus \set x$ is also a region of $M$. | From the definition, a region is a non-empty, open, path-connected subset of $M$.
First, note that as $R$ is open it can not be a singleton from Finite Subspace of Dense-in-itself Metric Space is not Open.
Therefore $R \setminus \set x$ is not empty.
Next, we see that from Open Set Less One Point is Open that $R$ is op... | Let $M = \struct {A, d}$ be a [[Definition:Dense-in-itself|dense-in-itself]] [[Definition:Metric Space|metric space]].
Let $R \subseteq M$ be a [[Definition:Region of Metric Space|region]] of $M$.
Let $x \in R$.
Then $R \setminus \set x$ is also a [[Definition:Region of Metric Space|region]] of $M$. | From the definition, a [[Definition:Region of Metric Space|region]] is a [[Definition:Empty Set|non-empty]], [[Definition:Open Set (Metric Space)|open]], [[Definition:Path-Connected|path-connected]] subset of $M$.
First, note that as $R$ is [[Definition:Open Set (Metric Space)|open]] it can not be a [[Definition:Sing... | Region Less One Point is Region | https://proofwiki.org/wiki/Region_Less_One_Point_is_Region | https://proofwiki.org/wiki/Region_Less_One_Point_is_Region | [
"Metric Spaces"
] | [
"Definition:Dense-in-itself",
"Definition:Metric Space",
"Definition:Region/Metric Space",
"Definition:Region/Metric Space"
] | [
"Definition:Region/Metric Space",
"Definition:Empty Set",
"Definition:Open Set/Metric Space",
"Definition:Path-Connected",
"Definition:Open Set/Metric Space",
"Definition:Singleton",
"Finite Subspace of Dense-in-itself Metric Space is not Open",
"Definition:Empty Set",
"Open Set Less One Point is Op... |
proofwiki-1967 | Theorem of Even Perfect Numbers | Let $a \in \N$ be an even perfect number.
Then $a$ is in the form:
:$2^{n - 1} \paren {2^n - 1}$
where $2^n - 1$ is prime.
Similarly, if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect. | === Sufficient Condition ===
{{:Theorem of Even Perfect Numbers/Sufficient Condition}}
{{qed|lemma}} | Let $a \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
Then $a$ is in the form:
:$2^{n - 1} \paren {2^n - 1}$
where $2^n - 1$ is [[Definition:Prime Number|prime]].
Similarly, if $2^n - 1$ is [[Definition:Prime Number|prime]], then $2^{n - 1} \paren {2^n - 1}$ is [[Definit... | === [[Theorem of Even Perfect Numbers/Sufficient Condition|Sufficient Condition]] ===
{{:Theorem of Even Perfect Numbers/Sufficient Condition}}
{{qed|lemma}} | Theorem of Even Perfect Numbers | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers | [
"Theorem of Even Perfect Numbers",
"Euclidean Numbers",
"Perfect Numbers",
"Mersenne Numbers",
"Mersenne Primes",
"Number Theory",
"Named Theorems"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Perfect Number"
] | [
"Theorem of Even Perfect Numbers/Sufficient Condition"
] |
proofwiki-1968 | Composite Number has Prime Factor not Greater Than its Square Root | Let $n \in \N$ and $n = p_1 \times p_2 \times \cdots \times p_j$, $j \ge 2$, where $p_1, \ldots, p_j \in \Bbb P$ are prime factors of $n$.
Then $\exists p_i \in \Bbb P$ such that $p_i \le \sqrt n$.
That is, if $n \in \N$ is composite, then $n$ has a prime factor $p \le \sqrt n$. | Let $n$ be composite such that $n \ge 0$.
From Composite Number has Two Divisors Less Than It, we can write $n = a b$ where $a, b \in \Z$ and $1 < a, b < n$.
{{WLOG}}, suppose that $a \le b$.
Let $a > \sqrt n$.
Then $b \ge a > \sqrt n$.
However, if $b \ge a > \sqrt n$ is true, then:
:$n = a b > \sqrt n \sqrt n > n$
T... | Let $n \in \N$ and $n = p_1 \times p_2 \times \cdots \times p_j$, $j \ge 2$, where $p_1, \ldots, p_j \in \Bbb P$ are [[Definition:Prime Factor|prime factors]] of $n$.
Then $\exists p_i \in \Bbb P$ such that $p_i \le \sqrt n$.
That is, if $n \in \N$ is [[Definition:Composite Number|composite]], then $n$ has a [[Defin... | Let $n$ be composite such that $n \ge 0$.
From [[Composite Number has Two Divisors Less Than It]], we can write $n = a b$ where $a, b \in \Z$ and $1 < a, b < n$.
{{WLOG}}, suppose that $a \le b$.
Let $a > \sqrt n$.
Then $b \ge a > \sqrt n$.
However, if $b \ge a > \sqrt n$ is true, then:
:$n = a b > \sqrt n \sqr... | Composite Number has Prime Factor not Greater Than its Square Root | https://proofwiki.org/wiki/Composite_Number_has_Prime_Factor_not_Greater_Than_its_Square_Root | https://proofwiki.org/wiki/Composite_Number_has_Prime_Factor_not_Greater_Than_its_Square_Root | [
"Prime Numbers"
] | [
"Definition:Prime Factor",
"Definition:Composite Number",
"Definition:Prime Factor"
] | [
"Composite Number has Two Divisors Less Than It",
"Definition:Contradiction",
"Positive Integer Greater than 1 has Prime Divisor",
"Definition:Prime Number",
"Absolute Value of Integer is not less than Divisors",
"Divisor Relation on Positive Integers is Partial Ordering"
] |
proofwiki-1969 | Lucas-Lehmer Test | Let $q$ be an odd prime.
Let $\sequence {L_n}_{n \mathop \in \N}$ be the recursive sequence in $\Z / \paren {2^q - 1} \Z$ defined by:
:$L_0 = 4, L_{n + 1} = L_n^2 - 2 \pmod {2^q - 1}$
Then $2^q - 1$ is prime {{iff}} $L_{q - 2} = 0 \pmod {2^q - 1}$. | Consider the sequences:
:$U_0 = 0, U_1 = 1, U_{n + 1} = 4 U_n - U_{n - 1}$
:$V_0 = 2, V_1 = 4, V_{n + 1} = 4 V_n - V_{n - 1}$
The following equations can be proved by induction:
{{begin-eqn}}
{{eqn | n = 1
| l = V_n
| r = U_{n + 1} - U_{n - 1}
| c =
}}
{{eqn | n = 2
| l = U_n
| r = \frac ... | Let $q$ be an [[Definition:Odd Integer|odd]] [[Definition:Prime Number|prime]].
Let $\sequence {L_n}_{n \mathop \in \N}$ be the [[Definition:Recursive Sequence|recursive sequence]] in $\Z / \paren {2^q - 1} \Z$ defined by:
:$L_0 = 4, L_{n + 1} = L_n^2 - 2 \pmod {2^q - 1}$
Then $2^q - 1$ is [[Definition:Prime Number|... | Consider the sequences:
:$U_0 = 0, U_1 = 1, U_{n + 1} = 4 U_n - U_{n - 1}$
:$V_0 = 2, V_1 = 4, V_{n + 1} = 4 V_n - V_{n - 1}$
The following equations can be proved by [[Principle of Mathematical Induction|induction]]:
{{begin-eqn}}
{{eqn | n = 1
| l = V_n
| r = U_{n + 1} - U_{n - 1}
| c =
}}
{{eqn... | Lucas-Lehmer Test | https://proofwiki.org/wiki/Lucas-Lehmer_Test | https://proofwiki.org/wiki/Lucas-Lehmer_Test | [
"Mersenne Numbers",
"Mersenne Primes",
"Prime Numbers"
] | [
"Definition:Odd Integer",
"Definition:Prime Number",
"Definition:Recursive Sequence",
"Definition:Prime Number"
] | [
"Principle of Mathematical Induction",
"Definition:Prime Number",
"Binomial Theorem",
"Definition:Odd Prime",
"Binomial Coefficient of Prime",
"Fermat's Little Theorem",
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Congruence (Number Theory)",
"Principle of Mathematical Inducti... |
proofwiki-1970 | Sum Over Divisors Equals Sum Over Quotients | Let $n$ be a positive integer.
Let $f: \Z_{>0} \to \Z_{>0}$ be a mapping on the positive integers.
Let $\ds \sum_{d \mathop \divides n} \map f d$ be the sum of $\map f d$ over the divisors of $n$.
Then:
:$\ds \sum_{d \mathop \divides n} \map f d = \sum_{d \mathop \divides n} \map f {\frac n d}$. | If $d$ is a divisor of $n$ then $d \times \dfrac n d = n$ and so $\dfrac n d$ is also a divisor of $n$.
Therefore if $d_1, d_2, \ldots, d_r$ are all the divisors of $n$, then so are $\dfrac n {d_1}, \dfrac n {d_2}, \ldots, \dfrac n {d_r}$, except in a different order.
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{d \mathop \d... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $f: \Z_{>0} \to \Z_{>0}$ be a [[Definition:Mapping|mapping]] on the [[Definition:Positive Integer|positive integers]].
Let $\ds \sum_{d \mathop \divides n} \map f d$ be the [[Definition:Sum Over Divisors|sum of $\map f d$ over the divisors of $n$]].
... | If $d$ is a [[Definition:Divisor of Integer|divisor]] of $n$ then $d \times \dfrac n d = n$ and so $\dfrac n d$ is also a [[Definition:Divisor of Integer|divisor]] of $n$.
Therefore if $d_1, d_2, \ldots, d_r$ are all the [[Definition:Divisor of Integer|divisors]] of $n$, then so are $\dfrac n {d_1}, \dfrac n {d_2}, \l... | Sum Over Divisors Equals Sum Over Quotients | https://proofwiki.org/wiki/Sum_Over_Divisors_Equals_Sum_Over_Quotients | https://proofwiki.org/wiki/Sum_Over_Divisors_Equals_Sum_Over_Quotients | [
"Number Theory",
"Divisors"
] | [
"Definition:Positive/Integer",
"Definition:Mapping",
"Definition:Positive/Integer",
"Definition:Sum Over Divisors"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Category:Number Theory",
"Category:Divisors"
] |
proofwiki-1971 | Sum of Reciprocals of Divisors equals Abundancy Index | Let $n$ be a positive integer.
Let $\map {\sigma_1} n$ denote the divisor sum function of $n$.
Then:
:$\ds \sum_{d \mathop \divides n} \frac 1 d = \frac {\map {\sigma_1} n} n$
where $\dfrac {\map {\sigma_1} n} n$ is the abundancy index of $n$. | {{begin-eqn}}
{{eqn | l = \sum_{d \mathop \divides n} \frac 1 d
| r = \sum_{d \mathop \divides n} \frac 1 {\paren {\frac n d} }
| c = Sum Over Divisors Equals Sum Over Quotients
}}
{{eqn | r = \frac 1 n \sum_{d \mathop \divides n} d
| c =
}}
{{eqn | r = \frac {\map {\sigma_1} n} n
| c = {{Defof... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $\map {\sigma_1} n$ denote the [[Definition:Divisor Sum Function|divisor sum function]] of $n$.
Then:
:$\ds \sum_{d \mathop \divides n} \frac 1 d = \frac {\map {\sigma_1} n} n$
where $\dfrac {\map {\sigma_1} n} n$ is the [[Definition:Abundancy Index|... | {{begin-eqn}}
{{eqn | l = \sum_{d \mathop \divides n} \frac 1 d
| r = \sum_{d \mathop \divides n} \frac 1 {\paren {\frac n d} }
| c = [[Sum Over Divisors Equals Sum Over Quotients]]
}}
{{eqn | r = \frac 1 n \sum_{d \mathop \divides n} d
| c =
}}
{{eqn | r = \frac {\map {\sigma_1} n} n
| c = {{D... | Sum of Reciprocals of Divisors equals Abundancy Index | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Divisors_equals_Abundancy_Index | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Divisors_equals_Abundancy_Index | [
"Number Theory",
"Abundancy"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Abundancy Index"
] | [
"Sum Over Divisors Equals Sum Over Quotients",
"Category:Number Theory",
"Category:Abundancy"
] |
proofwiki-1972 | Divisor Sum of Prime Number | Let $n$ be a positive integer.
Let $\map {\sigma_1} n$ be the divisor sum function of $n$.
Then $\map {\sigma_1} n = n + 1$ {{iff}} $n$ is prime. | From Rule of Transposition, we may replace the ''only if'' statement by its contrapositive.
Therefore, the following suffices: | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $\map {\sigma_1} n$ be the [[Definition:Divisor Sum Function|divisor sum function]] of $n$.
Then $\map {\sigma_1} n = n + 1$ {{iff}} $n$ is [[Definition:Prime Number|prime]]. | From [[Rule of Transposition]], we may replace the ''only if'' statement by its [[Definition:Contrapositive Statement|contrapositive]].
Therefore, the following suffices: | Divisor Sum of Prime Number | https://proofwiki.org/wiki/Divisor_Sum_of_Prime_Number | https://proofwiki.org/wiki/Divisor_Sum_of_Prime_Number | [
"Prime Numbers",
"Divisor Sum Function",
"Divisor Sum of Prime Number"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Prime Number"
] | [
"Rule of Transposition",
"Definition:Contrapositive Statement"
] |
proofwiki-1973 | Divisor Sum of Power of Prime | Let $n = p^k$ be the power of a prime number $p$.
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\map {\sigma_1} n = \dfrac {p^{k + 1} - 1} {p - 1}$ | From Divisors of Power of Prime, the divisors of $n = p^k$ are $1, p, p^2, \ldots, p^{k - 1}, p^k$.
Hence from Sum of Geometric Sequence:
:$\map {\sigma_1} {p^k} = 1 + p + p^2 + \cdots + p^{k - 1} + p^k = \dfrac {p^{k + 1} - 1} {p - 1}$
{{qed}} | Let $n = p^k$ be the [[Definition:Power (Algebra)|power]] of a [[Definition:Prime Number|prime number]] $p$.
Let $\map {\sigma_1} n$ be the [[Definition:Divisor Sum Function|divisor sum]] of $n$.
That is, let $\map {\sigma_1} n$ be the [[Definition:Integer Addition|sum]] of all [[Definition:Positive Integer|positive]... | From [[Divisors of Power of Prime]], the [[Definition:Divisor of Integer|divisors]] of $n = p^k$ are $1, p, p^2, \ldots, p^{k - 1}, p^k$.
Hence from [[Sum of Geometric Sequence]]:
:$\map {\sigma_1} {p^k} = 1 + p + p^2 + \cdots + p^{k - 1} + p^k = \dfrac {p^{k + 1} - 1} {p - 1}$
{{qed}} | Divisor Sum of Power of Prime | https://proofwiki.org/wiki/Divisor_Sum_of_Power_of_Prime | https://proofwiki.org/wiki/Divisor_Sum_of_Power_of_Prime | [
"Prime Numbers",
"Divisor Sum Function",
"Divisor Sum of Power of Prime"
] | [
"Definition:Power (Algebra)",
"Definition:Prime Number",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisors of Power of Prime",
"Definition:Divisor (Algebra)/Integer",
"Sum of Geometric Sequence"
] |
proofwiki-1974 | Divisor Count Function of Power of Prime | Let $n = p^k$ be the power of a prime number $p$.
Let $\map {\sigma_0} n$ be the divisor count function of $n$.
That is, let $\map {\sigma_0} n$ be the number of positive divisors of $n$.
Then:
:$\map {\sigma_0} n = k + 1$ | From Divisors of Power of Prime, the divisors of $n = p^k$ are:
:$1, p, p^2, \ldots, p^{k - 1}, p^k$
There are $k + 1$ of them.
Hence the result.
{{qed}} | Let $n = p^k$ be the [[Definition:Power (Algebra)|power]] of a [[Definition:Prime Number|prime number]] $p$.
Let $\map {\sigma_0} n$ be the [[Definition:Divisor Count Function|divisor count function]] of $n$.
That is, let $\map {\sigma_0} n$ be the number of positive [[Definition:Divisor of Integer|divisors]] of $n$.... | From [[Divisors of Power of Prime]], the [[Definition:Divisor of Integer|divisors]] of $n = p^k$ are:
:$1, p, p^2, \ldots, p^{k - 1}, p^k$
There are $k + 1$ of them.
Hence the result.
{{qed}} | Divisor Count Function of Power of Prime | https://proofwiki.org/wiki/Divisor_Count_Function_of_Power_of_Prime | https://proofwiki.org/wiki/Divisor_Count_Function_of_Power_of_Prime | [
"Divisor Count Function",
"Prime Numbers"
] | [
"Definition:Power (Algebra)",
"Definition:Prime Number",
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisors of Power of Prime",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-1975 | Sum Over Divisors of Multiplicative Function | Let $f: \Z_{>0} \to \Z_{>0}$ be a multiplicative function.
Let $n \in \Z_{>0}$.
Let $\ds \sum_{d \mathop \divides n} \map f d$ be the sum over the divisors of $n$.
Then $\ds \map F n = \sum_{d \mathop \divides n} \map f d$ is also a multiplicative function. | Let $\ds \map F n = \sum_{d \mathop \divides n} \map f d$.
Let $m, n \in \Z_{>0}: m \perp n$.
Then by definition:
:$\ds \map F {m n} = \sum_{d \mathop \divides m n} \map f d$
The divisors of $m n$ are of the form $d = r s$ where $r$ and $s$ are divisors of $m$ and $n$ respectively, from Divisors of Product of Coprime I... | Let $f: \Z_{>0} \to \Z_{>0}$ be a [[Definition:Multiplicative Arithmetic Function|multiplicative function]].
Let $n \in \Z_{>0}$.
Let $\ds \sum_{d \mathop \divides n} \map f d$ be the [[Definition:Sum Over Divisors|sum over the divisors]] of $n$.
Then $\ds \map F n = \sum_{d \mathop \divides n} \map f d$ is also a ... | Let $\ds \map F n = \sum_{d \mathop \divides n} \map f d$.
Let $m, n \in \Z_{>0}: m \perp n$.
Then by definition:
:$\ds \map F {m n} = \sum_{d \mathop \divides m n} \map f d$
The [[Definition:Divisor of Integer|divisors]] of $m n$ are of the form $d = r s$ where $r$ and $s$ are [[Definition:Divisor of Integer|diviso... | Sum Over Divisors of Multiplicative Function | https://proofwiki.org/wiki/Sum_Over_Divisors_of_Multiplicative_Function | https://proofwiki.org/wiki/Sum_Over_Divisors_of_Multiplicative_Function | [
"Multiplicative Functions"
] | [
"Definition:Multiplicative Arithmetic Function",
"Definition:Sum Over Divisors",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Divisors of Product of Coprime Integers",
"Definition:Common Divisor/Integers",
"Definition:Common Divisor/Integers",
"Definition:Multiplicative Arithmetic Function"
] |
proofwiki-1976 | Divisors of Product of Coprime Integers | Let $a \divides b c$, where $b \perp c$.
Then $\tuple {r, s}$ satisfying:
:$a = r s$, where $r \divides b$ and $s \divides c$
is unique up to absolute value with:
:$\size r = \gcd \set {a, b}$
:$\size s = \gcd \set {a, c}$ | By Divisor of Product, there exists $\tuple {r, s}$ satisfying:
:$r \divides b$
:$s \divides c$
:$r s = a$
We have:
:$r, s \divides a$
By definition of GCD:
:$\gcd \set {a, b} \divides r$
:$\gcd \set {a, c} \divides s$
By Absolute Value of Integer is not less than Divisors:
:$\gcd \set {a, b} \le \size r$
:$\gcd \set {... | Let $a \divides b c$, where $b \perp c$.
Then $\tuple {r, s}$ satisfying:
:$a = r s$, where $r \divides b$ and $s \divides c$
is [[Definition:Unique|unique]] up to [[Definition:Absolute Value|absolute value]] with:
:$\size r = \gcd \set {a, b}$
:$\size s = \gcd \set {a, c}$ | By [[Divisor of Product]], there exists $\tuple {r, s}$ satisfying:
:$r \divides b$
:$s \divides c$
:$r s = a$
We have:
:$r, s \divides a$
By definition of [[Definition:Greatest Common Divisor of Integers|GCD]]:
:$\gcd \set {a, b} \divides r$
:$\gcd \set {a, c} \divides s$
By [[Absolute Value of Integer is not less... | Divisors of Product of Coprime Integers | https://proofwiki.org/wiki/Divisors_of_Product_of_Coprime_Integers | https://proofwiki.org/wiki/Divisors_of_Product_of_Coprime_Integers | [
"Coprime Integers"
] | [
"Definition:Unique",
"Definition:Absolute Value"
] | [
"Divisor of Product",
"Definition:Greatest Common Divisor/Integers",
"Absolute Value of Integer is not less than Divisors",
"GCD with One Fixed Argument is Multiplicative Function",
"GCD of Integer and Divisor",
"Definition:Unique",
"Definition:Absolute Value",
"Category:Coprime Integers"
] |
proofwiki-1977 | Unity Function is Completely Multiplicative | Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the constant function:
:$\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$
Then $f_1$ is completely multiplicative. | :$\forall m, n \in \Z_{> 0}: f_1 \left({m n}\right) = 1 = f_1 \left({m}\right) f_1 \left({n}\right)$
{{qed}}
Category:Completely Multiplicative Functions
mhowtsfcrt0p94l8q38papewimpdmo0 | Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the [[Definition:Constant Mapping|constant function]]:
:$\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$
Then $f_1$ is [[Definition:Completely Multiplicative Function|completely multiplicative]]. | :$\forall m, n \in \Z_{> 0}: f_1 \left({m n}\right) = 1 = f_1 \left({m}\right) f_1 \left({n}\right)$
{{qed}}
[[Category:Completely Multiplicative Functions]]
mhowtsfcrt0p94l8q38papewimpdmo0 | Unity Function is Completely Multiplicative | https://proofwiki.org/wiki/Unity_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Unity_Function_is_Completely_Multiplicative | [
"Completely Multiplicative Functions"
] | [
"Definition:Constant Mapping",
"Definition:Completely Multiplicative Function"
] | [
"Category:Completely Multiplicative Functions"
] |
proofwiki-1978 | Identity Function is Completely Multiplicative | Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function:
:$\forall n \in \Z_{>0}: I_{\Z_{>0}} \left({n}\right) = n$
Then $I_{\Z_{>0}}$ is completely multiplicative.
{{Questionable|Completely multiplicativity is defined for fields but $\Z_{>0}$ is not a field}} | :$\forall m, n \in \Z_{>0}: I_{\Z_{>0}} \left({m n}\right) = m n = I_{\Z_{>0}} \left({m}\right) I_{\Z_{>0}} \left({n}\right)$
{{qed}}
Category:Completely Multiplicative Functions
Category:Identity Mappings
6ye9ue17l3wvccd3re221wv600xe3tm | Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Identity Mapping|identity function]]:
:$\forall n \in \Z_{>0}: I_{\Z_{>0}} \left({n}\right) = n$
Then $I_{\Z_{>0}}$ is [[Definition:Completely Multiplicative Function|completely multiplicative]].
{{Questionable|Completely multiplicativity is defined for fields... | :$\forall m, n \in \Z_{>0}: I_{\Z_{>0}} \left({m n}\right) = m n = I_{\Z_{>0}} \left({m}\right) I_{\Z_{>0}} \left({n}\right)$
{{qed}}
[[Category:Completely Multiplicative Functions]]
[[Category:Identity Mappings]]
6ye9ue17l3wvccd3re221wv600xe3tm | Identity Function is Completely Multiplicative | https://proofwiki.org/wiki/Identity_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Identity_Function_is_Completely_Multiplicative | [
"Completely Multiplicative Functions",
"Identity Mappings"
] | [
"Definition:Identity Mapping",
"Definition:Completely Multiplicative Function"
] | [
"Category:Completely Multiplicative Functions",
"Category:Identity Mappings"
] |
proofwiki-1979 | Divisor Count Function is Multiplicative | The divisor count function:
:$\ds \tau: \Z_{>0} \to \Z_{>0}: \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
is multiplicative. | Let $f_1: \Z_{>0} \to \Z_{>0}$ be the constant function:
:$\forall n \in \Z_{>0}: \map {f_1} n = 1$
Thus we have:
:$\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1 = \sum_{d \mathop \divides n} \map {f_1} d$
But from Unity Function is Completely Multiplicative, $f_1$ is multiplicative.
The result follows from Sum... | The [[Definition:Divisor Count Function|divisor count function]]:
:$\ds \tau: \Z_{>0} \to \Z_{>0}: \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
is [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | Let $f_1: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Constant Mapping|constant function]]:
:$\forall n \in \Z_{>0}: \map {f_1} n = 1$
Thus we have:
:$\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1 = \sum_{d \mathop \divides n} \map {f_1} d$
But from [[Unity Function is Completely Multiplicative]], $f_1$ is [[De... | Divisor Count Function is Multiplicative | https://proofwiki.org/wiki/Divisor_Count_Function_is_Multiplicative | https://proofwiki.org/wiki/Divisor_Count_Function_is_Multiplicative | [
"Multiplicative Functions",
"Divisor Count Function"
] | [
"Definition:Divisor Count Function",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Constant Mapping",
"Unity Function is Completely Multiplicative",
"Definition:Multiplicative Arithmetic Function",
"Sum Over Divisors of Multiplicative Function"
] |
proofwiki-1980 | Divisor Sum Function is Multiplicative | The divisor sum function:
:$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is multiplicative. | Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function:
:$\forall n \in \Z_{>0}: \map {I_{\Z_{>0} } } n = n$
Thus we have:
:$\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$
But from Identity Function is Completely Multiplicative, $I_{\Z_{>0} }$ is mul... | The [[Definition:Divisor Sum Function|divisor sum function]]:
:$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Identity Mapping|identity function]]:
:$\forall n \in \Z_{>0}: \map {I_{\Z_{>0} } } n = n$
Thus we have:
:$\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$
But from [[Identity Function is Completel... | Divisor Sum Function is Multiplicative/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_Function_is_Multiplicative | https://proofwiki.org/wiki/Divisor_Sum_Function_is_Multiplicative/Proof_1 | [
"Multiplicative Functions",
"Divisor Sum Function",
"Divisor Sum Function is Multiplicative"
] | [
"Definition:Divisor Sum Function",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Identity Mapping",
"Identity Function is Completely Multiplicative",
"Definition:Multiplicative Arithmetic Function",
"Sum Over Divisors of Multiplicative Function"
] |
proofwiki-1981 | Divisor Sum Function is Multiplicative | The divisor sum function:
:$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is multiplicative. | Let $a, b$ be coprime integers.
Because $a$ and $b$ have no common divisor, the divisors of $a b$ are integers of the form $a_i b_j$, where $a_i$ is a divisor of $a$ and $b_j$ is a divisor of $b$.
That is, any divisor $d$ of $a b$ is in the form:
:$d = a_i b_j$
in a unique way, where $a_i \divides a$ and $b_j \divides ... | The [[Definition:Divisor Sum Function|divisor sum function]]:
:$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | Let $a, b$ be [[Definition:Coprime Integers|coprime integers]].
Because $a$ and $b$ have no [[Definition:Common Divisor of Integers|common divisor]], the [[Definition:Divisor of Integer|divisors]] of $a b$ are [[Definition:Integer|integers]] of the form $a_i b_j$, where $a_i$ is a [[Definition:Divisor of Integer|divis... | Divisor Sum Function is Multiplicative/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_Function_is_Multiplicative | https://proofwiki.org/wiki/Divisor_Sum_Function_is_Multiplicative/Proof_2 | [
"Multiplicative Functions",
"Divisor Sum Function",
"Divisor Sum Function is Multiplicative"
] | [
"Definition:Divisor Sum Function",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Coprime/Integers",
"Definition:Common Divisor/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unique",
"Definition:Divisor (Algebra... |
proofwiki-1982 | Primes of form Power of Two plus One | Let $n \in \N$ be a natural number.
Let $2^n + 1$ be prime.
Then $n = 2^k$ for some natural number $k$. | Suppose $n$ has an odd divisor apart from $1$.
Then $n$ can be expressed as $n = \paren {2 r + 1} s$.
So:
{{begin-eqn}}
{{eqn | l = 2^n + 1
| r = 2^{\paren {2 r + 1} s} + 1
| c =
}}
{{eqn | r = \paren {2^s}^{\paren {2 r + 1} } + 1^{\paren {2 r + 1} }
| c =
}}
{{eqn | r = \paren {2^s + 1} \paren {2^{... | Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
Let $2^n + 1$ be [[Definition:Prime Number|prime]].
Then $n = 2^k$ for some [[Definition:Natural Numbers|natural number]] $k$. | Suppose $n$ has an [[Definition:Odd Integer|odd]] [[Definition:Divisor of Integer|divisor]] apart from $1$.
Then $n$ can be expressed as $n = \paren {2 r + 1} s$.
So:
{{begin-eqn}}
{{eqn | l = 2^n + 1
| r = 2^{\paren {2 r + 1} s} + 1
| c =
}}
{{eqn | r = \paren {2^s}^{\paren {2 r + 1} } + 1^{\paren {2 r ... | Primes of form Power of Two plus One/Proof 1 | https://proofwiki.org/wiki/Primes_of_form_Power_of_Two_plus_One | https://proofwiki.org/wiki/Primes_of_form_Power_of_Two_plus_One/Proof_1 | [
"Prime Numbers",
"Fermat Numbers",
"Primes of form Power of Two plus One"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Natural Numbers"
] | [
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Sum of Odd Positive Powers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Natural Numbers"
] |
proofwiki-1983 | Primes of form Power of Two plus One | Let $n \in \N$ be a natural number.
Let $2^n + 1$ be prime.
Then $n = 2^k$ for some natural number $k$. | A specific instance of Primes of form Power plus One:
$q^n + 1$ is prime only if:
:$(1): \quad q$ is even
and
:$(2): \quad n$ is of the form $2^k$ for some positive integer $k$.
As $2$ is even, the result applies.
{{qed}} | Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
Let $2^n + 1$ be [[Definition:Prime Number|prime]].
Then $n = 2^k$ for some [[Definition:Natural Numbers|natural number]] $k$. | A specific instance of [[Primes of form Power plus One]]:
$q^n + 1$ is [[Definition:Prime Number|prime]] only if:
:$(1): \quad q$ is [[Definition:Even Integer|even]]
and
:$(2): \quad n$ is of the form $2^k$ for some [[Definition:Positive Integer|positive integer]] $k$.
As $2$ is [[Definition:Even Integer|even]], the... | Primes of form Power of Two plus One/Proof 2 | https://proofwiki.org/wiki/Primes_of_form_Power_of_Two_plus_One | https://proofwiki.org/wiki/Primes_of_form_Power_of_Two_plus_One/Proof_2 | [
"Prime Numbers",
"Fermat Numbers",
"Primes of form Power of Two plus One"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Natural Numbers"
] | [
"Primes of form Power plus One",
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Positive/Integer",
"Definition:Even Integer"
] |
proofwiki-1984 | Integer has Multiplicative Order Modulo n iff Coprime to n | Let $a$ and $n$ be integers.
Let the multiplicative order of $a$ modulo $n$ exist.
Then:
:$a \perp n$
that is, $a$ and $n$ are coprime. | === Necessary Condition ===
Suppose $c \in \Z_{>0}$ is the multiplicative order of $a$ modulo $n$.
Then by definition:
:$a^c \equiv 1 \pmod n$
Hence, by definition:
:$a^c = k n + 1$
for some $k \in \Z$.
Thus:
:$a r + n s = 1$
where:
:$r = a^{c - 1}$
:$s = -k$
It follows from Integer Combination of Coprime Integers that... | Let $a$ and $n$ be [[Definition:Integer|integers]].
Let the [[Definition:Multiplicative Order of Integer|multiplicative order of $a$ modulo $n$]] exist.
Then:
:$a \perp n$
that is, $a$ and $n$ are [[Definition:Coprime Integers|coprime]]. | === Necessary Condition ===
Suppose $c \in \Z_{>0}$ is the [[Definition:Multiplicative Order of Integer|multiplicative order of $a$ modulo $n$]].
Then by definition:
:$a^c \equiv 1 \pmod n$
Hence, by definition:
:$a^c = k n + 1$
for some $k \in \Z$.
Thus:
:$a r + n s = 1$
where:
:$r = a^{c - 1}$
:$s = -k$
It fo... | Integer has Multiplicative Order Modulo n iff Coprime to n | https://proofwiki.org/wiki/Integer_has_Multiplicative_Order_Modulo_n_iff_Coprime_to_n | https://proofwiki.org/wiki/Integer_has_Multiplicative_Order_Modulo_n_iff_Coprime_to_n | [
"Coprime Integers",
"Modulo Arithmetic"
] | [
"Definition:Integer",
"Definition:Multiplicative Order of Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Multiplicative Order of Integer",
"Integer Combination of Coprime Integers",
"Definition:Coprime/Integers",
"Definition:Multiplicative Order of Integer"
] |
proofwiki-1985 | Integer to Power of Multiple of Order | Let $a$ and $n$ be integers.
Let $a \perp n$, that is, let $a$ and $b$ be coprime.
Let $c \in \Z_{>0}$ be the multiplicative order of $a$ modulo $n$.
Then $a^k \equiv 1 \pmod n$ {{iff}} $k$ is a multiple of $c$. | First note from Integer has Multiplicative Order Modulo n iff Coprime to n that unless $a \perp n$ the multiplicative order of $a$ modulo $n$ does not exist. | Let $a$ and $n$ be [[Definition:Integer|integers]].
Let $a \perp n$, that is, let $a$ and $b$ be [[Definition:Coprime Integers|coprime]].
Let $c \in \Z_{>0}$ be the [[Definition:Multiplicative Order of Integer|multiplicative order of $a$ modulo $n$]].
Then $a^k \equiv 1 \pmod n$ {{iff}} $k$ is a [[Definition:Multip... | First note from [[Integer has Multiplicative Order Modulo n iff Coprime to n]] that unless $a \perp n$ the [[Definition:Multiplicative Order of Integer|multiplicative order of $a$ modulo $n$]] does not exist. | Integer to Power of Multiple of Order | https://proofwiki.org/wiki/Integer_to_Power_of_Multiple_of_Order | https://proofwiki.org/wiki/Integer_to_Power_of_Multiple_of_Order | [
"Integer to Power of Multiple of Order",
"Number Theory"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Multiplicative Order of Integer",
"Definition:Multiple/Integer"
] | [
"Integer has Multiplicative Order Modulo n iff Coprime to n",
"Definition:Multiplicative Order of Integer"
] |
proofwiki-1986 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty z^n
| r = -z^0 + \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = -1 + \frac 1 {1 - z}
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \frac {z - 1 + 1} {1 - z}
| c =
}}
{{eqn | r = \frac z {1 - z}
| c =
}}
{{end-eqn}}
{{qed... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty z^n
| r = -z^0 + \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = -1 + \frac 1 {1 - z}
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \frac {z - 1 + 1} {1 - z}
| c =
}}
{{eqn | r = \frac z {1 - z}
| c =
}}
{{end-eqn}}
{... | Sum of Infinite Geometric Sequence/Corollary 1/Proof 1 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Corollary_1/Proof_1 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-1987 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty z^n
| r = \sum_{n \mathop = 1}^\infty z \cdot z^{n - 1}
| c =
}}
{{eqn | r = z \sum_{n \mathop = 1}^\infty z^{n - 1}
| c =
}}
{{eqn | r = z \sum_{m \mathop = 0}^\infty z^m
| c = setting $m = n - 1$
}}
{{eqn | r = z \frac 1 {1 - z}
| c =... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty z^n
| r = \sum_{n \mathop = 1}^\infty z \cdot z^{n - 1}
| c =
}}
{{eqn | r = z \sum_{n \mathop = 1}^\infty z^{n - 1}
| c =
}}
{{eqn | r = z \sum_{m \mathop = 0}^\infty z^m
| c = setting $m = n - 1$
}}
{{eqn | r = z \frac 1 {1 - z}
| c =... | Sum of Infinite Geometric Sequence/Corollary 1/Proof 2 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Corollary_1/Proof_2 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-1988 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | From Sum of Geometric Sequence, we have:
:$\ds s_N = \sum_{n \mathop = 0}^N z^n = \frac {1 - z^{N + 1} } {1 - z}$
We have that $\size z < 1$.
So by Sequence of Powers of Number less than One:
:$z^{N + 1} \to 0$ as $N \to \infty$
Hence $s_N \to \dfrac 1 {1 - z}$ as $N \to \infty$.
The result follows.
{{qed|lemma}}
It re... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | From [[Sum of Geometric Sequence]], we have:
:$\ds s_N = \sum_{n \mathop = 0}^N z^n = \frac {1 - z^{N + 1} } {1 - z}$
We have that $\size z < 1$.
So by [[Sequence of Powers of Number less than One]]:
:$z^{N + 1} \to 0$ as $N \to \infty$
Hence $s_N \to \dfrac 1 {1 - z}$ as $N \to \infty$.
The result follows.
{{qed|l... | Sum of Infinite Geometric Sequence/Proof 1 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Proof_1 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Sum of Geometric Sequence",
"Sequence of Powers of Number less than One",
"Definition:Absolutely Convergent Series",
"Definition:Absolute Value"
] |
proofwiki-1989 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | By the Chain Rule for Derivatives and {{Corollary|Nth Derivative of Reciprocal of Mth Power|disp = $n$th Derivative of Reciprocal of $m$th Power}}:
:$\dfrac {\d^n} {\d z^n} \dfrac 1 {1 - z} = \dfrac {n!} {\paren {1 - z}^{n + 1} }$
Thus the Maclaurin series expansion of $\dfrac 1 {1 - z}$ is:
:$\ds \sum_{n \mathop = 0}^... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | By the [[Chain Rule for Derivatives]] and {{Corollary|Nth Derivative of Reciprocal of Mth Power|disp = $n$th Derivative of Reciprocal of $m$th Power}}:
:$\dfrac {\d^n} {\d z^n} \dfrac 1 {1 - z} = \dfrac {n!} {\paren {1 - z}^{n + 1} }$
Thus the [[Definition:Maclaurin Series|Maclaurin series expansion]] of $\dfrac 1 {1 ... | Sum of Infinite Geometric Sequence/Proof 2 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Proof_2 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Derivative of Composite Function",
"Definition:Maclaurin Series"
] |
proofwiki-1990 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | Let $S = \ds \sum_{n \mathop = 0}^\infty z^n$.
Then:
{{begin-eqn}}
{{eqn | l = z S
| r = z \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty z^{n + 1}
| c =
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty z^n
| c = Translation of Index Variable of Summation
}}
{{eqn |... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | Let $S = \ds \sum_{n \mathop = 0}^\infty z^n$.
Then:
{{begin-eqn}}
{{eqn | l = z S
| r = z \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty z^{n + 1}
| c =
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty z^n
| c = [[Translation of Index Variable of Summation]]
}}
{... | Sum of Infinite Geometric Sequence/Proof 3 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Proof_3 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Translation of Index Variable of Summation"
] |
proofwiki-1991 | Sum of Infinite Geometric Sequence | Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the absolute value of $z$, for real and rational $z$
:the complex modulus of $z$ for complex $z$.
Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$. | {{begin-eqn}}
{{eqn | l = \frac 1 {1 - z}
| r = \frac 1 {1 + \paren {-z} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-z}^k
| c = Power Series Expansion for $\dfrac 1 {1 + z}$
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-1}^k z^k
| c =
}}
{{eqn ... | Let $S$ be a [[Definition:Standard Number Field|standard number field]], that is $\Q$, $\R$ or $\C$.
Let $z \in S$.
Let $\size z < 1$, where $\size z$ denotes:
:the [[Definition:Absolute Value|absolute value]] of $z$, for [[Definition:Real Number|real]] and [[Definition:Rational Number|rational]] $z$
:the [[Definiti... | {{begin-eqn}}
{{eqn | l = \frac 1 {1 - z}
| r = \frac 1 {1 + \paren {-z} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-z}^k
| c = [[Power Series Expansion for Reciprocal of 1 + x|Power Series Expansion for $\dfrac 1 {1 + z}$]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \pa... | Sum of Infinite Geometric Sequence/Proof 4 | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Geometric_Sequence/Proof_4 | [
"Sum of Infinite Geometric Sequence",
"Sum of Geometric Sequence",
"Sums of Sequences",
"Geometric Sequences",
"Convergence Tests",
"Series",
"Examples of Power Series"
] | [
"Definition:Standard Number Field",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Absolutely Convergent Series"
] | [
"Power Series Expansion for Reciprocal of 1 + x"
] |
proofwiki-1992 | Square of Riemann Zeta Function | :$\ds \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map {\sigma_0} k} {k^z}$
where:
:$\zeta$ is the Riemann zeta function
:$\sigma_0$ is the divisor count function. | {{begin-eqn}}
{{eqn | l = \map {\zeta^2} z
| r = \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} } \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} }
| c =
}}
{{eqn | r = \paren {1 + \frac 1 {2^z} + \frac 1 {3^z} + \frac 1 {4^z} + \frac 1 {5^z} + \frac 1 {6^z} + \cdots} \paren {1 + \frac 1 {2^z} + \frac... | :$\ds \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map {\sigma_0} k} {k^z}$
where:
:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann zeta function]]
:$\sigma_0$ is the [[Definition:Divisor Count Function|divisor count function]]. | {{begin-eqn}}
{{eqn | l = \map {\zeta^2} z
| r = \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} } \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} }
| c =
}}
{{eqn | r = \paren {1 + \frac 1 {2^z} + \frac 1 {3^z} + \frac 1 {4^z} + \frac 1 {5^z} + \frac 1 {6^z} + \cdots} \paren {1 + \frac 1 {2^z} + \frac... | Square of Riemann Zeta Function | https://proofwiki.org/wiki/Square_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Square_of_Riemann_Zeta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Divisor Count Function"
] | [
"Definition:Divisor (Algebra)/Integer",
"Category:Riemann Zeta Function"
] |
proofwiki-1993 | Derivative of Riemann Zeta Function | The derivative of the Riemann zeta function is:
:$\ds \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$ | {{begin-eqn}}
{{eqn | l = \frac {\d \zeta} {\d z}
| r = \map {\frac \d {\d z} } {\sum_{n \mathop = 1}^\infty n^{-z} }
| c = {{Defof|Riemann Zeta Function}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \map {\frac \d {\d z} } {n^{-z} }
| c = Sum Rule for Derivatives/General Result
}}
{{eqn | r = \sum_{n ... | The [[Definition:Derivative|derivative]] of the [[Definition:Riemann Zeta Function|Riemann zeta function]] is:
:$\ds \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$ | {{begin-eqn}}
{{eqn | l = \frac {\d \zeta} {\d z}
| r = \map {\frac \d {\d z} } {\sum_{n \mathop = 1}^\infty n^{-z} }
| c = {{Defof|Riemann Zeta Function}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \map {\frac \d {\d z} } {n^{-z} }
| c = [[Sum Rule for Derivatives/General Result]]
}}
{{eqn | r = \sum... | Derivative of Riemann Zeta Function | https://proofwiki.org/wiki/Derivative_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Derivative_of_Riemann_Zeta_Function | [
"Derivatives",
"Riemann Zeta Function"
] | [
"Definition:Derivative",
"Definition:Riemann Zeta Function"
] | [
"Sum Rule for Derivatives/General Result",
"Derivative of Exponential Function",
"Exponent Combination Laws/Negative Power",
"Natural Logarithm of 1 is 0"
] |
proofwiki-1994 | Order of Divisor Count Function | For all $x \ge 1$:
:$\ds \sum_{n \mathop \le x} \map {\sigma_0} n = x \log x + \paren {2 \gamma - 1} x + \map \OO {\sqrt x}$
where:
:$\gamma$ is the Euler-Mascheroni constant
:$\map {\sigma_0} n$ is the divisor count function. | {{ProofWanted}}
Category:Number Theory
swpkcthxo9hdv7mpwo3j0q8es7awl5q | For all $x \ge 1$:
:$\ds \sum_{n \mathop \le x} \map {\sigma_0} n = x \log x + \paren {2 \gamma - 1} x + \map \OO {\sqrt x}$
where:
:$\gamma$ is the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]
:$\map {\sigma_0} n$ is the [[Definition:Divisor Count Function|divisor count function]]. | {{ProofWanted}}
[[Category:Number Theory]]
swpkcthxo9hdv7mpwo3j0q8es7awl5q | Order of Divisor Count Function | https://proofwiki.org/wiki/Order_of_Divisor_Count_Function | https://proofwiki.org/wiki/Order_of_Divisor_Count_Function | [
"Number Theory"
] | [
"Definition:Euler-Mascheroni Constant",
"Definition:Divisor Count Function"
] | [
"Category:Number Theory"
] |
proofwiki-1995 | Order of Möbius Function | Let $\mu$ denote the Möbius function .
Then:
:$\ds \sum_{n \mathop \le N} \map \mu n = \map o N$
where $o$ denotes little-o notation. | Let $\map \Re z$ be the real part of a complex variable $z$.
By Dirichlet Series is Analytic, the Riemann zeta function is analytic.
By Trivial Zeroes of Riemann Zeta Function are Even Negative Integers, the Riemann zeta function has no zeroes in $\map \Re z > 1$.
Thus the reciprocal of the Riemann zeta function is ana... | Let $\mu$ denote the [[Definition:Möbius Function|Möbius function]] .
Then:
:$\ds \sum_{n \mathop \le N} \map \mu n = \map o N$
where $o$ denotes [[Definition:Little-O Notation|little-o notation]]. | Let $\map \Re z$ be the [[Definition:Real Part|real part]] of a [[Definition:Complex Variable|complex variable]] $z$.
By [[Dirichlet Series is Analytic]], the [[Definition:Riemann Zeta Function|Riemann zeta function]] is [[Definition:Analytic Function|analytic]].
By [[Trivial Zeroes of Riemann Zeta Function are Even ... | Order of Möbius Function | https://proofwiki.org/wiki/Order_of_Möbius_Function | https://proofwiki.org/wiki/Order_of_Möbius_Function | [
"Möbius Function",
"Analytic Number Theory"
] | [
"Definition:Möbius Function",
"Definition:Little-O Notation"
] | [
"Definition:Complex Number/Real Part",
"Definition:Variable/Complex",
"Dirichlet Series is Analytic",
"Definition:Riemann Zeta Function",
"Definition:Analytic Function",
"Trivial Zeroes of Riemann Zeta Function are Even Negative Integers",
"Definition:Riemann Zeta Function",
"Definition:Root of Mappin... |
proofwiki-1996 | Ingham's Theorem on Convergent Dirichlet Series | Let $\sequence {a_n} \le 1$
{{explain|What exactly is $a_n$ in this context?}}
For a complex number $z \in \C$, let $\map \Re z$ denote the real part of $z$.
Form the series $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z}$ which converges to an analytic function $\map F z$ for $\map \Re z > 1$.
{{explain|We have "$\map \R... | Fix a $w$ in $\map \Re w \ge 1$.
Then $\map F {z + w}$ is analytic in $\map \Re z \ge 0$.
{{explain|The above needs to be proved. We know that $\map F z$ is analytic throughout $\map \Re z \ge 1$ but we are told nothing about what it's like on $\map \Re z \ge 0$. Also see below where it is also stated that it is analyt... | Let $\sequence {a_n} \le 1$
{{explain|What exactly is $a_n$ in this context?}}
For a [[Definition:Complex Number|complex number]] $z \in \C$, let $\map \Re z$ denote the [[Definition:Real Part|real part]] of $z$.
Form the series $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z}$ which converges to an [[Definition:Analyti... | Fix a $w$ in $\map \Re w \ge 1$.
Then $\map F {z + w}$ is analytic in $\map \Re z \ge 0$.
{{explain|The above needs to be proved. We know that $\map F z$ is analytic throughout $\map \Re z \ge 1$ but we are told nothing about what it's like on $\map \Re z \ge 0$. Also see below where it is also stated that it is anal... | Ingham's Theorem on Convergent Dirichlet Series | https://proofwiki.org/wiki/Ingham's_Theorem_on_Convergent_Dirichlet_Series | https://proofwiki.org/wiki/Ingham's_Theorem_on_Convergent_Dirichlet_Series | [
"Dirichlet Series"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Analytic Function"
] | [
"Definition:Curve/Arc",
"Cauchy's Residue Theorem",
"Cauchy's Residue Theorem",
"Category:Dirichlet Series"
] |
proofwiki-1997 | Alexander Polynomial is a Knot Invariant | The Alexander polynomial of an elementary knot $K$ is invariant under Reidemeister moves. | {{ProofWanted}}
Category:Knot Theory
cfk3l9xnijecc9vcqsutu3c1dsb9y4m | The [[Definition:Alexander Polynomial|Alexander polynomial]] of an [[Definition:Elementary Knot|elementary knot]] $K$ is invariant under [[Definition:Reidemeister Move|Reidemeister moves]]. | {{ProofWanted}}
[[Category:Knot Theory]]
cfk3l9xnijecc9vcqsutu3c1dsb9y4m | Alexander Polynomial is a Knot Invariant | https://proofwiki.org/wiki/Alexander_Polynomial_is_a_Knot_Invariant | https://proofwiki.org/wiki/Alexander_Polynomial_is_a_Knot_Invariant | [
"Knot Theory"
] | [
"Definition:Alexander Polynomial",
"Definition:Knot (Knot Theory)/Elementary Knot",
"Definition:Reidemeister Move"
] | [
"Category:Knot Theory"
] |
proofwiki-1998 | Product Form of Sum on Completely Multiplicative Function | Let $f$ be a completely multiplicative arithmetic function.
Let the series $\ds \sum_{n \mathop = 1}^\infty \map f n$ be absolutely convergent.
Then:
:$\ds \sum_{n \mathop = 1}^\infty \map f n = \prod_p \frac 1 {1 - \map f p}$
where the infinite product ranges over the primes. | Define $P$ by:
{{begin-eqn}}
{{eqn | l = \map P {A, K}
| o = :=
| r = \prod_{\substack {p \mathop \in \mathbb P \\ p \mathop \le A} } \frac {1 - \map f p^{K + 1} } {1 - \map f p}
| c = where $\mathbb P$ denotes the set of prime numbers
}}
{{eqn | r = \prod_{\substack {p \mathop \in \mathbb P \\ p \mat... | Let $f$ be a [[Definition:Completely Multiplicative Function|completely multiplicative]] [[Definition:Arithmetic Function|arithmetic function]].
Let the [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\infty \map f n$ be [[Definition:Absolutely Convergent Series|absolutely convergent]].
Then:
:$\ds \sum_{n \m... | Define $P$ by:
{{begin-eqn}}
{{eqn | l = \map P {A, K}
| o = :=
| r = \prod_{\substack {p \mathop \in \mathbb P \\ p \mathop \le A} } \frac {1 - \map f p^{K + 1} } {1 - \map f p}
| c = where $\mathbb P$ denotes the [[Definition:Set|set]] of [[Definition:Prime Number|prime numbers]]
}}
{{eqn | r = \pro... | Product Form of Sum on Completely Multiplicative Function | https://proofwiki.org/wiki/Product_Form_of_Sum_on_Completely_Multiplicative_Function | https://proofwiki.org/wiki/Product_Form_of_Sum_on_Completely_Multiplicative_Function | [
"Completely Multiplicative Functions"
] | [
"Definition:Completely Multiplicative Function",
"Definition:Arithmetic Function",
"Definition:Series",
"Definition:Absolutely Convergent Series",
"Definition:Continued Product/Infinite",
"Definition:Prime Number"
] | [
"Definition:Set",
"Definition:Prime Number",
"Sum of Geometric Sequence",
"Product of Summations is Summation Over Cartesian Product of Products",
"Definition:Completely Multiplicative Function",
"Fundamental Theorem of Arithmetic",
"Definition:Convergence",
"Definition:Absolutely Convergent Series"
] |
proofwiki-1999 | Power Function is Completely Multiplicative | Let $K$ be a field.
Let $z \in K$.
Let $f_z: K \to K$ be the mapping defined as:
:$\forall x \in K: \map {f_z} x = x^z$
Then $f_z$ is completely multiplicative. | Let $r, s \in K$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f_z} {r s}
| r = \paren {r s}^z
| c = Definition of $f_z$
}}
{{eqn | r = \paren r^z \paren s^z
| c = Power of Product in Abelian Group
}}
{{eqn | r = \map {f_z} r \map {f_z} s
| c = Definition of $f_z$
}}
{{end-eqn}}
{{Qed}}
Category:Field ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $z \in K$.
Let $f_z: K \to K$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in K: \map {f_z} x = x^z$
Then $f_z$ is [[Definition:Completely Multiplicative Function|completely multiplicative]]. | Let $r, s \in K$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f_z} {r s}
| r = \paren {r s}^z
| c = Definition of $f_z$
}}
{{eqn | r = \paren r^z \paren s^z
| c = [[Power of Product in Abelian Group]]
}}
{{eqn | r = \map {f_z} r \map {f_z} s
| c = Definition of $f_z$
}}
{{end-eqn}}
{{Qed}}
[[Catego... | Power Function is Completely Multiplicative | https://proofwiki.org/wiki/Power_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Power_Function_is_Completely_Multiplicative | [
"Field Theory",
"Completely Multiplicative Functions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Mapping",
"Definition:Completely Multiplicative Function"
] | [
"Power of Product in Abelian Group",
"Category:Field Theory",
"Category:Completely Multiplicative Functions"
] |
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