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-20700 | Field Generated by Macroscopic Charge Density | Let $B$ be a body of matter.
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The electric field at $P$ generated by the macroscopic charge density within $B$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf... | From Electric Field Strength from Assemblage of Point Charges, the electric field strength caused by an assemblage of point charges $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {\paren {\mathbf r - \mathbf r_i} q_i} {\size {\mathbf r - \mathbf r_i}^... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Matter|matter]].
Let $P$ be a [[Definition:Point|point]] inside $B$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The [[Definition:Electric Field|electric field]] at $P$ generated by the [[Definition:Macroscopic Charge Density|macroscopic c... | From [[Electric Field Strength from Assemblage of Point Charges]], the [[Definition:Electric Field Strength|electric field strength]] caused by an assemblage of [[Definition:Point Charge|point charges]] $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {... | Field Generated by Macroscopic Charge Density | https://proofwiki.org/wiki/Field_Generated_by_Macroscopic_Charge_Density | https://proofwiki.org/wiki/Field_Generated_by_Macroscopic_Charge_Density | [
"Macroscopic Charge Density"
] | [
"Definition:Body",
"Definition:Matter",
"Definition:Point",
"Definition:Position Vector",
"Definition:Electric Field",
"Definition:Macroscopic Charge Density",
"Definition:Infinitesimal",
"Definition:Volume Element",
"Definition:Position Vector",
"Definition:Macroscopic Charge Density",
"Definit... | [
"Electric Field Strength from Assemblage of Point Charges",
"Definition:Electric Field Strength",
"Definition:Point Charge",
"Definition:Position Vector",
"Definition:Macroscopic Charge Density",
"Definition:Summation",
"Definition:Definite Integral",
"Definition:Volume Element",
"Definition:Scale o... |
proofwiki-20701 | Field Generated by Surface Charge Density | Let $B$ be a body of matter.
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The electric field at $P$ generated by the surface charge density over $B$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'... | From Electric Field Strength from Assemblage of Point Charges, the electric field strength caused by an assemblage of point charges $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {\paren {\mathbf r - \mathbf r_i} q_i} {\size {\mathbf r - \mathbf r_i}^... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Matter|matter]].
Let $P$ be a [[Definition:Point|point]] inside $B$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The [[Definition:Electric Field|electric field]] at $P$ generated by the [[Definition:Surface Charge Density|surface charge de... | From [[Electric Field Strength from Assemblage of Point Charges]], the [[Definition:Electric Field Strength|electric field strength]] caused by an assemblage of [[Definition:Point Charge|point charges]] $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {... | Field Generated by Surface Charge Density | https://proofwiki.org/wiki/Field_Generated_by_Surface_Charge_Density | https://proofwiki.org/wiki/Field_Generated_by_Surface_Charge_Density | [
"Field Generated by Surface Charge Density",
"Surface Charge Density"
] | [
"Definition:Body",
"Definition:Matter",
"Definition:Point",
"Definition:Position Vector",
"Definition:Electric Field",
"Definition:Surface Charge Density",
"Definition:Infinitesimal",
"Definition:Area Element",
"Definition:Position Vector",
"Definition:Surface Charge Density",
"Definition:Vacuum... | [
"Electric Field Strength from Assemblage of Point Charges",
"Definition:Electric Field Strength",
"Definition:Point Charge",
"Definition:Position Vector",
"Definition:Surface Charge Density",
"Definition:Summation",
"Definition:Definite Integral",
"Definition:Area Element",
"Definition:Electric Fiel... |
proofwiki-20702 | Macroscopic Electric Field in Body | Let $B$ be a body of matter.
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The macroscopic electric field at $P$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\math... | From Field Generated by Macroscopic Charge Density, the electric field at $P$ generated by the macroscopic charge density within $B$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Matter|matter]].
Let $P$ be a [[Definition:Point|point]] inside $B$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The [[Definition:Macroscopic Electric Field|macroscopic electric field]] at $P$ is given by:
:$\ds \map {\mathbf E} {\mathbf... | From [[Field Generated by Macroscopic Charge Density]], the [[Definition:Electric Field|electric field]] at $P$ generated by the [[Definition:Macroscopic Charge Density|macroscopic charge density]] within $B$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \df... | Macroscopic Electric Field in Body | https://proofwiki.org/wiki/Macroscopic_Electric_Field_in_Body | https://proofwiki.org/wiki/Macroscopic_Electric_Field_in_Body | [
"Macroscopic Electric Fields"
] | [
"Definition:Body",
"Definition:Matter",
"Definition:Point",
"Definition:Position Vector",
"Definition:Macroscopic Electric Field",
"Definition:Infinitesimal",
"Definition:Volume Element",
"Definition:Infinitesimal",
"Definition:Area Element",
"Definition:Position Vector",
"Definition:Macroscopic... | [
"Field Generated by Macroscopic Charge Density",
"Definition:Electric Field",
"Definition:Macroscopic Charge Density",
"Field Generated by Surface Charge Density",
"Definition:Electric Field",
"Definition:Surface Charge Density",
"Principle of Superposition/Examples/Electric Field",
"Definition:Macros... |
proofwiki-20703 | Characterization of Open Set by Open Cover | Let $T = \struct{S, \tau}$ be a topological space.
Let $\UU$ be an open cover of $T$.
For each $U \in \UU$, let $\tau_U$ denote the subspace topology on $U$.
Let $W \subseteq S$.
Then $W$ is open in $T$ {{iff}}:
:$\forall U \in \UU: W \cap U$ is open in $\struct{U, \tau_U}$ | === Necessary Condition ===
This follows immediately from the definition of subspace topology.
{{qed|lemma}} | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
For each $U \in \UU$, let $\tau_U$ denote the [[Definition:Subspace Topology|subspace topology]] on $U$.
Let $W \subseteq S$.
Then $W$ is [[Definition:Open Set (Topolog... | === Necessary Condition ===
This follows immediately from the definition of [[Definition:Subspace Topology|subspace topology]].
{{qed|lemma}} | Characterization of Open Set by Open Cover | https://proofwiki.org/wiki/Characterization_of_Open_Set_by_Open_Cover | https://proofwiki.org/wiki/Characterization_of_Open_Set_by_Open_Cover | [
"Open Sets",
"Covers"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Topological Subspace"
] |
proofwiki-20704 | Characterization of Closed Set by Open Cover | Let $T = \struct{S, \tau}$ be a topological space.
Let $\UU$ be an open cover of $T$.
For each $U \in \UU$, let $\tau_U$ denote the subspace topology on $U$.
Let $F \subseteq S$.
Then $F$ is closed in $T$ {{iff}}:
:$\forall U \in \UU: F \cap U$ is closed in $\struct{U, \tau_U}$ | === Necessary Condition ===
This follows immediately from Closed Set in Topological Subspace.
{{qed|lemma}} | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
For each $U \in \UU$, let $\tau_U$ denote the [[Definition:Subspace Topology|subspace topology]] on $U$.
Let $F \subseteq S$.
Then $F$ is [[Definition:Closed Set (Topol... | === Necessary Condition ===
This follows immediately from [[Closed Set in Topological Subspace]].
{{qed|lemma}} | Characterization of Closed Set by Open Cover | https://proofwiki.org/wiki/Characterization_of_Closed_Set_by_Open_Cover | https://proofwiki.org/wiki/Characterization_of_Closed_Set_by_Open_Cover | [
"Closed Sets",
"Covers"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Closed Set in Topological Subspace"
] |
proofwiki-20705 | Union of Closed Locally Finite Set of Subsets is Closed | Let $T = \struct{S, \tau}$ be a topological space.
Let $\FF$ be a closed locally finite set of subsets of $T$.
Let $E = \ds \bigcup \FF$.
Then:
:$E$ is closed in $T$. | Let:
:$\UU = \leftset {U \in \tau : \set {F \in \FF : U \cap F \ne \O}}$ is finite $\rightset{}$
By definition of closed locally finite set of subsets:
:$\forall x \in S : \exists U \in \tau : x \in U : \set{F \in \FF : U \cap F \ne \O}$ is finite
That is:
:$\forall x \in S : \exists U \in \UU : x \in U$
Hence
:$\UU$ i... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite set of subsets]] of $T$.
Let $E = \ds \bigcup \FF$.
Then:
:$E$ is [[Definition:Closed Set (Topology)|closed]] in $T$. | Let:
:$\UU = \leftset {U \in \tau : \set {F \in \FF : U \cap F \ne \O}}$ is [[Definition:Finite Set|finite]] $\rightset{}$
By definition of [[Definition:Closed Locally Finite Set of Subsets|closed locally finite set of subsets]]:
:$\forall x \in S : \exists U \in \tau : x \in U : \set{F \in \FF : U \cap F \ne \O}$ is... | Union of Closed Locally Finite Set of Subsets is Closed | https://proofwiki.org/wiki/Union_of_Closed_Locally_Finite_Set_of_Subsets_is_Closed | https://proofwiki.org/wiki/Union_of_Closed_Locally_Finite_Set_of_Subsets_is_Closed | [
"Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Closed Set/Topology"
] | [
"Definition:Finite Set",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Finite Set",
"Definition:Open Cover",
"Intersection Distributes over Union",
"Union with Empty Set",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"D... |
proofwiki-20706 | Closures of Elements of Locally Finite Set is Locally Finite | Let $T = \struct{S, \tau}$ be a topological space.
Let $\AA$ be a locally finite set of subsets of $T$.
Then:
:$\set{A^- : A \in \AA}$ is locally finite in $T$
where $A^-$ denotes the closure of $A$ in $T$. | Let $\BB = \set{A^- : A \in \AA}$.
Let $x \in S$.
By definition of locally finite set of subsets:
:$\exists U_x \in \tau : x \in U_x :$ the set $\AA_x = \set{A \in \AA : A \cap U_x \ne \O}$ is finite
Let $\BB_x = \set{A^- \in \BB : A^- \cap U_x \ne \O}$
Let $A^- \in \BB_x$.
From Open Set Disjoint from Set is Disjoint f... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\AA$ be a [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]] of $T$.
Then:
:$\set{A^- : A \in \AA}$ is [[Definition:Locally Finite Set of Subsets|locally finite]] in $T$
where $A^-$ denotes the [[Definit... | Let $\BB = \set{A^- : A \in \AA}$.
Let $x \in S$.
By definition of [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]]:
:$\exists U_x \in \tau : x \in U_x :$ the [[Definition:Set|set]] $\AA_x = \set{A \in \AA : A \cap U_x \ne \O}$ is [[Definition:Finite Set|finite]]
Let $\BB_x = \set{A^- \in... | Closures of Elements of Locally Finite Set is Locally Finite | https://proofwiki.org/wiki/Closures_of_Elements_of_Locally_Finite_Set_is_Locally_Finite | https://proofwiki.org/wiki/Closures_of_Elements_of_Locally_Finite_Set_is_Locally_Finite | [
"Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Definition:Closure (Topology)"
] | [
"Definition:Locally Finite Set of Subsets",
"Definition:Set",
"Definition:Finite Set",
"Open Set Disjoint from Set is Disjoint from Closure",
"Definition:Finite Set",
"Definition:Set",
"Definition:Finite Set",
"Definition:Locally Finite Set of Subsets",
"Category:Locally Finite Sets of Subsets"
] |
proofwiki-20707 | Union of Closures of Elements of Locally Finite Set is Closed | Let $T = \struct{S, \tau}$ be a topological space.
Let $\AA$ be a locally finite set of subsets of $T$.
Then:
:$\ds \paren {\bigcup \AA}^- = \bigcup \set{A^- : A \in \AA}$
where $A^-$ denotes the closure of $A$ in $T$. | From Closures of Elements of Locally Finite Set is Locally Finite:
:$\set{A^- : A \in \AA}$ is also locally finite
From Union of Closed Locally Finite Set of Subsets is Closed:
:$\bigcup \set{A^- : A \in \AA}$ is closed in $T$
We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \AA
| l = A
| o = \subseteq
... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\AA$ be a [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]] of $T$.
Then:
:$\ds \paren {\bigcup \AA}^- = \bigcup \set{A^- : A \in \AA}$
where $A^-$ denotes the [[Definition:Closure (Topology)|closure]] ... | From [[Closures of Elements of Locally Finite Set is Locally Finite]]:
:$\set{A^- : A \in \AA}$ is also [[Definition:Locally Finite Set of Subsets|locally finite]]
From [[Union of Closed Locally Finite Set of Subsets is Closed]]:
:$\bigcup \set{A^- : A \in \AA}$ is [[Definition:Closed Set (Topology)|closed]] in $T$
... | Union of Closures of Elements of Locally Finite Set is Closed | https://proofwiki.org/wiki/Union_of_Closures_of_Elements_of_Locally_Finite_Set_is_Closed | https://proofwiki.org/wiki/Union_of_Closures_of_Elements_of_Locally_Finite_Set_is_Closed | [
"Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Locally Finite Set of Subsets",
"Definition:Closure (Topology)"
] | [
"Closures of Elements of Locally Finite Set is Locally Finite",
"Definition:Locally Finite Set of Subsets",
"Union of Closed Locally Finite Set of Subsets is Closed",
"Definition:Closed Set/Topology",
"Set is Subset of its Topological Closure",
"Set is Subset of Union",
"Union of Subsets is Subset",
"... |
proofwiki-20708 | Electric Flux out of Closed Surface surrounding Point Charge | Let $q$ be a point charge.
Let $S$ be a closed surface surrounding $q$.
The total electric flux through $S$ is given by:
:$F = \dfrac q {\varepsilon_0}$ | === Lemma ===
{{:Electric Flux out of Closed Surface surrounding Point Charge/Lemma}}{{qed|lemma}}
From this lemma, we can transform the surface integral for the flux through the entire surface $S$ into a double integral between angles:
{{begin-eqn}}
{{eqn | l = \int_S \mathbf E \cdot \rd \mathbf S
| r = \dfrac q... | Let $q$ be a [[Definition:Point Charge|point charge]].
Let $S$ be a [[Definition:Closed Surface|closed surface]] surrounding $q$.
The total [[Definition:Electric Flux|electric flux]] through $S$ is given by:
:$F = \dfrac q {\varepsilon_0}$ | === [[Electric Flux out of Closed Surface surrounding Point Charge/Lemma|Lemma]] ===
{{:Electric Flux out of Closed Surface surrounding Point Charge/Lemma}}{{qed|lemma}}
From this [[Electric Flux out of Closed Surface surrounding Point Charge/Lemma|lemma]], we can transform the [[Definition:Surface Integral|surface i... | Electric Flux out of Closed Surface surrounding Point Charge | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Point_Charge | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Point_Charge | [
"Electric Flux out of Closed Surface",
"Electric Flux",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Closed Surface",
"Definition:Electric Flux"
] | [
"Electric Flux out of Closed Surface surrounding Point Charge/Lemma",
"Electric Flux out of Closed Surface surrounding Point Charge/Lemma",
"Definition:Surface Integral",
"Definition:Electric Flux",
"Definition:Surface",
"Definition:Multiple Integral/Double",
"Definition:Angle"
] |
proofwiki-20709 | Electric Flux out of Closed Surface surrounding Point Charge/Lemma | Let $q$ be a point charge located at the origin of a spherical polar coordinate system.
{{DefinitionWanted|Spherical Polar Coordinate System}}
Let $S$ be a closed surface surrounding $q$.
Let $\delta \mathbf S$ be an area element of $S$.
Let $\delta \Omega$ be the solid angle subtended by the projection of $\delta \mat... | Consider the area element $\delta \mathbf S$ at the position $P$ with position vector $\mathbf r$ whose spherical coordinates are $\polar {r, \theta, \phi}$.
Let $\delta S_p$ be the projected area of $\delta \mathbf S$ to the plane perpendicular to $\mathbf r$.
:500px
Let us define $\delta \mathbf S$ as being the subse... | Let $q$ be a [[Definition:Point Charge|point charge]] located at the [[Definition:Origin|origin]] of a [[Definition:Spherical Polar Coordinate System|spherical polar coordinate system]].
{{DefinitionWanted|Spherical Polar Coordinate System}}
Let $S$ be a [[Definition:Closed Surface|closed surface]] surrounding $q$.
... | Consider the [[Definition:Area Element|area element]] $\delta \mathbf S$ at the [[Definition:Position|position]] $P$ with [[Definition:Position Vector|position vector]] $\mathbf r$ whose [[Definition:Spherical Polar Coordinates|spherical coordinates]] are $\polar {r, \theta, \phi}$.
Let $\delta S_p$ be the [[Definitio... | Electric Flux out of Closed Surface surrounding Point Charge/Lemma | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Point_Charge/Lemma | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Point_Charge/Lemma | [
"Electric Flux out of Closed Surface",
"Electric Flux",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Coordinate System/Origin",
"Definition:Spherical Polar Coordinate System",
"Definition:Closed Surface",
"Definition:Area Element",
"Definition:Solid Angle Subtended ",
"Definition:Projection (Geometry)",
"Definition:Electric Flux"
] | [
"Definition:Area Element",
"Definition:Position",
"Definition:Position Vector",
"Definition:Spherical Coordinate System",
"Definition:Projection (Geometry)",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"File:Area-Element-Spherical.png",
"Definition:Subset",
"Definitio... |
proofwiki-20710 | Electric Flux out of Closed Surface surrounding Assemblage of Point Charges | Let $Q = \set {q_1, q_2, \ldots}$ be a set of point charges.
Let $S$ be a closed surface surrounding $Q$.
The total electric flux through $S$ is given by:
:$\ds F = \dfrac 1 {\varepsilon_0} \sum_Q q_i$ | From Electric Flux out of Closed Surface surrounding Point Charge:
:$F_i = \dfrac {q_i} {\varepsilon_0}$
where $F_i$ is the part of $F$ brought about by $q_i$.
The result follows from Electric Field satisfies Principle of Superposition.
{{qed}} | Let $Q = \set {q_1, q_2, \ldots}$ be a [[Definition:Set|set]] of [[Definition:Point Charge|point charges]].
Let $S$ be a [[Definition:Closed Surface|closed surface]] surrounding $Q$.
The total [[Definition:Electric Flux|electric flux]] through $S$ is given by:
:$\ds F = \dfrac 1 {\varepsilon_0} \sum_Q q_i$ | From [[Electric Flux out of Closed Surface surrounding Point Charge]]:
:$F_i = \dfrac {q_i} {\varepsilon_0}$
where $F_i$ is the part of $F$ brought about by $q_i$.
The result follows from [[Electric Field satisfies Principle of Superposition]].
{{qed}} | Electric Flux out of Closed Surface surrounding Assemblage of Point Charges | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Assemblage_of_Point_Charges | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Assemblage_of_Point_Charges | [
"Electric Flux out of Closed Surface",
"Electric Flux",
"Point Charges"
] | [
"Definition:Set",
"Definition:Point Charge",
"Definition:Closed Surface",
"Definition:Electric Flux"
] | [
"Electric Flux out of Closed Surface surrounding Point Charge",
"Principle of Superposition/Examples/Electric Field"
] |
proofwiki-20711 | Electric Flux out of Closed Surface surrounding Body with Continuous Charge Distribution | Let $B$ be a body of matter which has a continuous macroscopic charge density $\map \rho {\mathbf r}$.
Let $S$ be a closed surface surrounding $Q$.
The total electric flux through $S$ generated by the electric charge on $B$ is given by:
:$\ds F = \dfrac 1 {\varepsilon_0} \int_V \map \rho {\mathbf r} \rd \tau$
where:
:$... | Let $\d \tau$ be an arbitrary infinitesimal volume element.
$\d \tau$ can be considered as an infinitesimal point charge $\d q$.
From Electric Flux out of Closed Surface surrounding Point Charge:
:$\d F_\tau = \dfrac {\d q} {\varepsilon_0}$
where $\d F_\tau$ is the infinitesimal part of $F$ brought about by $\d q$.
By... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Matter|matter]] which has a [[Definition:Continuous Real-Valued Vector Function|continuous]] [[Definition:Macroscopic Charge Density|macroscopic charge density]] $\map \rho {\mathbf r}$.
Let $S$ be a [[Definition:Closed Surface|closed surface]] surrounding $Q$.
Th... | Let $\d \tau$ be an arbitrary [[Definition:Infinitesimal|infinitesimal]] [[Definition:Volume Element|volume element]].
$\d \tau$ can be considered as an [[Definition:Infinitesimal|infinitesimal]] [[Definition:Point Charge|point charge]] $\d q$.
From [[Electric Flux out of Closed Surface surrounding Point Charge]]:
:... | Electric Flux out of Closed Surface surrounding Body with Continuous Charge Distribution | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Body_with_Continuous_Charge_Distribution | https://proofwiki.org/wiki/Electric_Flux_out_of_Closed_Surface_surrounding_Body_with_Continuous_Charge_Distribution | [
"Electric Flux out of Closed Surface",
"Electric Flux",
"Point Charges"
] | [
"Definition:Body",
"Definition:Matter",
"Definition:Continuous Real-Valued Vector Function",
"Definition:Macroscopic Charge Density",
"Definition:Closed Surface",
"Definition:Electric Flux",
"Definition:Electric Charge",
"Definition:Volume",
"Definition:Infinitesimal",
"Definition:Volume Element",... | [
"Definition:Infinitesimal",
"Definition:Volume Element",
"Definition:Infinitesimal",
"Definition:Point Charge",
"Electric Flux out of Closed Surface surrounding Point Charge",
"Definition:Infinitesimal",
"Definition:Macroscopic Charge Density",
"Definition:Definite Integral",
"Definition:Ordinary Sp... |
proofwiki-20712 | Rotation of Cartesian Axes around Vector | Let $\mathbf r$ be a vector in space.
Let a Cartesian plane $\CC$ be established such that:
:the initial point of $\mathbf r$ is at the origin $O$
:the terminal point of $\mathbf r$ is the point $P$.
Let $\tuple {X, Y}$ be the coordinates of $P$ under $\CC$.
Let $\CC$ be rotated about $O$ to $\CC'$, through an angle $\... | :500px
With reference to the above diagram:
:$X P X' = \varphi$
and so:
:$OX' = OX \cos \varphi + PX \sin \varphi$
and:
:$OY' = OY \cos \varphi - PY \cos \varphi$
Hence the result.
{{qed}} | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector]] in [[Definition:Ordinary Space|space]].
Let a [[Definition:Cartesian Plane|Cartesian plane]] $\CC$ be established such that:
:the [[Definition:Initial Point of Vector|initial point]] of $\mathbf r$ is at the [[Definition:Origin|origin]] $O$
:the [[Definition:T... | :[[File:Rotation-of-Cartesian-Plane-around-Vector.png|500px]]
With reference to the above diagram:
:$X P X' = \varphi$
and so:
:$OX' = OX \cos \varphi + PX \sin \varphi$
and:
:$OY' = OY \cos \varphi - PY \cos \varphi$
Hence the result.
{{qed}} | Rotation of Cartesian Axes around Vector | https://proofwiki.org/wiki/Rotation_of_Cartesian_Axes_around_Vector | https://proofwiki.org/wiki/Rotation_of_Cartesian_Axes_around_Vector | [
"Cartesian Coordinate Systems",
"Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Ordinary Space",
"Definition:Cartesian Plane",
"Definition:Initial Point of Vector",
"Definition:Coordinate System/Origin",
"Definition:Terminal Point of Vector",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Rotation (Geometry)/P... | [
"File:Rotation-of-Cartesian-Plane-around-Vector.png"
] |
proofwiki-20713 | Hex Theorem | A Hex board in which every tile is marked has exactly one winner. | {{ProofWanted}}
Category:Hex Theorem
Category:Hex (Game)
mqwpdq7b61fgqddeqg26n5q0l8p9vmx | A [[Definition:Hex Board|Hex board]] in which every [[Definition:Tile of Hex Board|tile]] is marked has exactly one [[Definition:Hex Winner|winner]]. | {{ProofWanted}}
[[Category:Hex Theorem]]
[[Category:Hex (Game)]]
mqwpdq7b61fgqddeqg26n5q0l8p9vmx | Hex Theorem | https://proofwiki.org/wiki/Hex_Theorem | https://proofwiki.org/wiki/Hex_Theorem | [
"Hex Theorem",
"Hex (Game)"
] | [
"Definition:Hex (Game)/Board",
"Definition:Hex (Game)/Board/Tile",
"Definition:Hex (Game)/Winning"
] | [
"Category:Hex Theorem",
"Category:Hex (Game)"
] |
proofwiki-20714 | T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1 | Let $F$ be a closed subset of $T$.
Let $X \subseteq S$ be disjoint from $F$.
Then there exists a countable open cover $\WW = \set{W_n : n \in \N}$ of $X$:
:$\forall n \in \N : W_n^- \cap F = \O$ | Let $\UU = \set {U \in \BB : U^- \cap F = \O}$. | Let $F$ be a [[Definition:Closed Subset|closed subset]] of $T$.
Let $X \subseteq S$ be [[Definition:Disjoint Sets|disjoint]] from $F$.
Then there exists a [[Definition:Countable Set|countable]] [[Definition:Open Cover|open cover]] $\WW = \set{W_n : n \in \N}$ of $X$:
:$\forall n \in \N : W_n^- \cap F = \O$ | Let $\UU = \set {U \in \BB : U^- \cap F = \O}$. | T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Lemma_1 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Lemma_1 | [
"T3 Space with Sigma-Locally Finite Basis is T4 Space"
] | [
"Definition:Closed Subset",
"Definition:Disjoint Sets",
"Definition:Countable Set",
"Definition:Open Cover"
] | [] |
proofwiki-20715 | T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 2 | Let $x \in S \setminus F$.
Then:
:$\exists U \in \BB : x \in U : U^- \cap F = \O$ | From Characterization of T3 Space:
:$\exists W \in \tau: x \in W : W^- \cap F = \O$
By definition of basis:
:$\exists U \in \BB : x \in U \subseteq W$
From Topological Closure of Subset is Subset of Topological Closure:
:$U^- \subseteq W^-$
From Set Intersection Preserves Subsets:
:$U^- \cap F = \O$
{{qed}}
Category:T3... | Let $x \in S \setminus F$.
Then:
:$\exists U \in \BB : x \in U : U^- \cap F = \O$ | From [[Characterization of T3 Space]]:
:$\exists W \in \tau: x \in W : W^- \cap F = \O$
By definition of [[Definition:Synthetic Basis|basis]]:
:$\exists U \in \BB : x \in U \subseteq W$
From [[Topological Closure of Subset is Subset of Topological Closure]]:
:$U^- \subseteq W^-$
From [[Set Intersection Preserves Sub... | T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 2 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Lemma_2 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Lemma_2 | [
"T3 Space with Sigma-Locally Finite Basis is T4 Space"
] | [] | [
"Characterization of T3 Space",
"Definition:Basis (Topology)/Synthetic Basis",
"Topological Closure of Subset is Subset of Topological Closure",
"Set Intersection Preserves Subsets",
"Category:T3 Space with Sigma-Locally Finite Basis is T4 Space"
] |
proofwiki-20716 | Countable Open Covers Condition for Separated Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B \subseteq S$
For all $X \subseteq S$, let $X^-$ denote the closure of $X$ in $T$.
Let:
:$\UU = \set {U_n : n \in \N}$ be a countable open cover of $A : \forall n \in \N : {U_n}^- \cap B = \O$
Let:
:$\VV = \set {V_n : n \in \N}$ be a countable open cover of $... | By definition of cover:
:$\ds A \subseteq \bigcup_{n \mathop \in \N} U_n$
We have:
:$\ds A \cap \paren{\bigcup_{n \mathop \in \N} {V_n}^-} = \O$
From Subset of Set Difference iff Disjoint Set:
:$(1): \quad \ds A \subseteq \paren {\bigcup_{n \mathop \in \N} U_n} \setminus \paren {\bigcup_{n \mathop \in \N} {V_n}^-}$
Sim... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B \subseteq S$
For all $X \subseteq S$, let $X^-$ denote the [[Definition:Topological Closure|closure]] of $X$ in $T$.
Let:
:$\UU = \set {U_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Open Cover... | By definition of [[Definition:Cover of Set|cover]]:
:$\ds A \subseteq \bigcup_{n \mathop \in \N} U_n$
We have:
:$\ds A \cap \paren{\bigcup_{n \mathop \in \N} {V_n}^-} = \O$
From [[Subset of Set Difference iff Disjoint Set]]:
:$(1): \quad \ds A \subseteq \paren {\bigcup_{n \mathop \in \N} U_n} \setminus \paren {\big... | Countable Open Covers Condition for Separated Sets | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets | [
"Separated Sets",
"Countable Open Covers Condition for Separated Sets"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Separated Sets"
] | [
"Definition:Cover of Set",
"Subset of Set Difference iff Disjoint Set"
] |
proofwiki-20717 | Countable Open Covers Condition for Separated Sets/Lemma 1 | :$\forall n, m \in \N : {U_n}' \cap {V_m}' = \O$ | Let $n, m \in \N$.
{{WLOG}}, let $m \le n$.
We have:
{{begin-eqn}}
{{eqn | l = {U_m}'
| r = U_m \setminus \paren {\bigcup_{p \mathop \le m} {V_p}^-}
| c = Definition of ${U_m}'$
}}
{{eqn | o = \subseteq
| r = U_m
| c = Set Difference is Subset
}}
{{eqn | o = \subseteq
| r = {U_m}^-
... | :$\forall n, m \in \N : {U_n}' \cap {V_m}' = \O$ | Let $n, m \in \N$.
{{WLOG}}, let $m \le n$.
We have:
{{begin-eqn}}
{{eqn | l = {U_m}'
| r = U_m \setminus \paren {\bigcup_{p \mathop \le m} {V_p}^-}
| c = Definition of ${U_m}'$
}}
{{eqn | o = \subseteq
| r = U_m
| c = [[Set Difference is Subset]]
}}
{{eqn | o = \subseteq
| r = {U_m}... | Countable Open Covers Condition for Separated Sets/Lemma 1 | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets/Lemma_1 | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets/Lemma_1 | [
"Countable Open Covers Condition for Separated Sets"
] | [] | [
"Set Difference is Subset",
"Set is Subset of its Topological Closure",
"Set is Subset of Union",
"Set Difference with Subset is Superset of Set Difference",
"Set Intersection Preserves Subsets",
"Set Difference Intersection with Second Set is Empty Set",
"Empty Set is Subset of All Sets",
"Definition... |
proofwiki-20718 | Countable Open Covers Condition for Separated Sets/Lemma 2 | :$U$ and $V$ are open in $T$. | By {{Open-set-axiom|2}}, it is sufficient to show that:
:$\forall n \in \N : {U_n}', {V_n}' \in \tau$
Let $n \in \N$.
We have:
{{begin-eqn}}
{{eqn | l = {U_n}'
| r = U_n \setminus \paren {\ds \bigcup_{p \mathop \le n} {V_p}^-}
| c = Definition of ${U_n}'$
}}
{{eqn | r = U_n \cap \relcomp S {\bigcup_{p \mat... | :$U$ and $V$ are [[Definition:Open Set (Topology)|open]] in $T$. | By {{Open-set-axiom|2}}, it is sufficient to show that:
:$\forall n \in \N : {U_n}', {V_n}' \in \tau$
Let $n \in \N$.
We have:
{{begin-eqn}}
{{eqn | l = {U_n}'
| r = U_n \setminus \paren {\ds \bigcup_{p \mathop \le n} {V_p}^-}
| c = Definition of ${U_n}'$
}}
{{eqn | r = U_n \cap \relcomp S {\bigcup_{p \m... | Countable Open Covers Condition for Separated Sets/Lemma 2 | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets/Lemma_2 | https://proofwiki.org/wiki/Countable_Open_Covers_Condition_for_Separated_Sets/Lemma_2 | [
"Countable Open Covers Condition for Separated Sets"
] | [
"Definition:Open Set/Topology"
] | [
"Set Difference as Intersection with Relative Complement",
"De Morgan's Laws (Set Theory)",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Set Intersection/Finite Intersection",
"Definition:Open Set/Topol... |
proofwiki-20719 | Square Function is Even | The square function:
:$\map f x = x^2$
is an even function. | {{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \paren {-x}^2
| r = \paren {-x} \times \paren {-x}
| c =
}}
{{eqn | r = \paren {-1}^2 x^2
| c =
}}
{{eqn | r = x^2
| c =
}}
{{end-eqn}}
Hence the result by definition of even function.
{{qed}} | The [[Definition:Square Function|square function]]:
:$\map f x = x^2$
is an [[Definition:Even Function|even function]]. | {{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \paren {-x}^2
| r = \paren {-x} \times \paren {-x}
| c =
}}
{{eqn | r = \paren {-1}^2 x^2
| c =
}}
{{eqn | r = x^2
| c =
}}
{{end-eqn}}
Hence the result by definition of [[Definition:Even Function|even function]].
{{qed}} | Square Function is Even | https://proofwiki.org/wiki/Square_Function_is_Even | https://proofwiki.org/wiki/Square_Function_is_Even | [
"Square Function",
"Examples of Even Functions"
] | [
"Definition:Square/Function",
"Definition:Even Function"
] | [
"Definition:Even Function"
] |
proofwiki-20720 | Subset of Locally Finite Set of Subsets is Locally Finite | Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is locally finite then $\GG$ is locally finite. | We prove the contrapositive statement:
:If $\GG$ is not locally finite then $\FF$ is not locally finite.
Let $\GG$ not be locally finite.
By definition of locally finite:
:$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : N$ intersects an infinite number of sets in $\GG$.
By definition of subset:
... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF \subseteq \powerset S$ be a [[Definition:Set of Sets|set of subsets]] of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is [[Definition:Locally Finite Set of Subsets|locally finite]] then $\GG$ is [[Definition:Locally Finite Set ... | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:If $\GG$ is not [[Definition:Locally Finite Set of Subsets|locally finite]] then $\FF$ is not [[Definition:Locally Finite Set of Subsets|locally finite]].
Let $\GG$ not be [[Definition:Locally Finite Set of Subsets|locally finite]].
By d... | Subset of Locally Finite Set of Subsets is Locally Finite | https://proofwiki.org/wiki/Subset_of_Locally_Finite_Set_of_Subsets_is_Locally_Finite | https://proofwiki.org/wiki/Subset_of_Locally_Finite_Set_of_Subsets_is_Locally_Finite | [
"Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets"
] | [
"Definition:Contrapositive Statement",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Definition:Neighborhood (Topology)",
"Definition:Set Intersection",
"Definition:Infini... |
proofwiki-20721 | Cube Function is Odd | The cube function on the real numbers:
:$\forall x \in \R: \map f x = x^3$
is an odd function. | {{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \paren {-x}^3
| r = \paren {-1}^3 x^3
| c =
}}
{{eqn | r = -x^3
| c =
}}
{{end-eqn}}
Hence the result by definition of odd function.
{{qed}} | The [[Definition:Cube (Algebra)|cube function]] on the [[Definition:Real Number|real numbers]]:
:$\forall x \in \R: \map f x = x^3$
is an [[Definition:Odd Function|odd function]]. | {{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \paren {-x}^3
| r = \paren {-1}^3 x^3
| c =
}}
{{eqn | r = -x^3
| c =
}}
{{end-eqn}}
Hence the result by definition of [[Definition:Odd Function|odd function]].
{{qed}} | Cube Function is Odd | https://proofwiki.org/wiki/Cube_Function_is_Odd | https://proofwiki.org/wiki/Cube_Function_is_Odd | [
"Examples of Odd Functions"
] | [
"Definition:Cube/Algebra",
"Definition:Real Number",
"Definition:Odd Function"
] | [
"Definition:Odd Function"
] |
proofwiki-20722 | Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite | Let $T = \struct{S, \tau}$ be a topological space.
Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is $\sigma$-locally finite then $\GG$ is $\sigma$-locally finite. | By definition of $\sigma$-locally finite:
:$\FF = \ds \bigcup_{n \in \N} \FF_n$
where $\FF_n$ is locally finite for each $n \in \N$.
For each $n \in \N$, let:
:$\GG_n = \GG \cap \FF_n$
We have:
{{begin-eqn}}
{{eqn | l = \GG
| r = \GG \cap \FF
| c = Intersection with Subset is Subset
}}
{{eqn | r = \GG \cap ... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF \subseteq \powerset S$ be a [[Definition:Set of Sets|set of subsets]] of $S$.
Let $\GG \subseteq \FF$.
If $\FF$ is [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] then $\GG$ is [[Definition:Sigma... | By definition of [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]]:
:$\FF = \ds \bigcup_{n \in \N} \FF_n$
where $\FF_n$ is [[Definition:Locally Finite Set of Subsets|locally finite]] for each $n \in \N$.
For each $n \in \N$, let:
:$\GG_n = \GG \cap \FF_n$
We have:
{{begin-eqn}}
{{eqn | l = \... | Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite | https://proofwiki.org/wiki/Subset_of_Sigma-Locally_Finite_Set_of_Subsets_is_Sigma-Locally_Finite | https://proofwiki.org/wiki/Subset_of_Sigma-Locally_Finite_Set_of_Subsets_is_Sigma-Locally_Finite | [
"Sigma-Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Sigma-Locally Finite Set of Subsets"
] | [
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Intersection with Subset is Subset",
"Intersection Distributes over Union",
"Intersection is Subset",
"Subset of Locally Finite Set of Subsets is Locally Finite",
"Definition:Locally Finite Set of Subsets",
... |
proofwiki-20723 | Real Function is Expressible as Sum of Even Function and Odd Function | Let $f: \R \to \R$ be a real function which is neither an even function nor an odd function.
Then $f$ may be expressed as the pointwise sum of an even function and an odd function. | Let:
{{begin-eqn}}
{{eqn | l = \map g x
| r = \dfrac {\map f x + \map f {-x} } 2
}}
{{eqn | l = \map h x
| r = \dfrac {\map f x - \map f {-x} } 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = \map g {-x}
| r = \dfrac {\map f {-x} + \map f {-\paren {-x} } } 2
| c =
}}
{{eqn | r = \dfrac... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is neither an [[Definition:Even Function|even function]] nor an [[Definition:Odd Function|odd function]].
Then $f$ may be expressed as the [[Definition:Pointwise Addition|pointwise sum]] of an [[Definition:Even Function|even function]] and an [[D... | Let:
{{begin-eqn}}
{{eqn | l = \map g x
| r = \dfrac {\map f x + \map f {-x} } 2
}}
{{eqn | l = \map h x
| r = \dfrac {\map f x - \map f {-x} } 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = \map g {-x}
| r = \dfrac {\map f {-x} + \map f {-\paren {-x} } } 2
| c =
}}
{{eqn | r = \df... | Real Function is Expressible as Sum of Even Function and Odd Function | https://proofwiki.org/wiki/Real_Function_is_Expressible_as_Sum_of_Even_Function_and_Odd_Function | https://proofwiki.org/wiki/Real_Function_is_Expressible_as_Sum_of_Even_Function_and_Odd_Function | [
"Real Function is Expressible as Sum of Even Function and Odd Function",
"Real Functions",
"Odd Functions",
"Even Functions"
] | [
"Definition:Real Function",
"Definition:Even Function",
"Definition:Odd Function",
"Definition:Pointwise Addition",
"Definition:Even Function",
"Definition:Odd Function"
] | [
"Definition:Even Function",
"Definition:Odd Function"
] |
proofwiki-20724 | Characteristics of Vector in Plane | Let a Cartesian plane $\CC$ be established with origin $O$.
Let $\tuple {A_x, A_y}$ be an ordered pair of real numbers that can be used to represent a point in $\CC$.
Then:
:$\tuple {A_x, A_y}$ are the Cartesian coordinates of the terminal point of a position vector $\mathbf A$
{{iff}}:
:$\tuple {A_x, A_y}$ can be tran... | === Sufficient Condition ===
Let $\tuple {A_x, A_y}$ represent the terminal point of a position vector $\mathbf A$.
Then from Rotation of Cartesian Axes around Vector:
{{begin-eqn}}
{{eqn | l = {A'}_x
| r = A_x \cos \varphi + A_y \sin \varphi
| c =
}}
{{eqn | l = {A'}_y
| r = -A_x \sin \varphi + A_y ... | Let a [[Definition:Cartesian Plane|Cartesian plane]] $\CC$ be established with [[Definition:Origin|origin]] $O$.
Let $\tuple {A_x, A_y}$ be an [[Definition:Ordered Pair|ordered pair]] of [[Definition:Real Number|real numbers]] that can be used to represent a [[Definition:Point|point]] in $\CC$.
Then:
:$\tuple {A_x, ... | === Sufficient Condition ===
Let $\tuple {A_x, A_y}$ represent the [[Definition:Terminal Point of Vector|terminal point]] of a [[Definition:Position Vector|position vector]] $\mathbf A$.
Then from [[Rotation of Cartesian Axes around Vector]]:
{{begin-eqn}}
{{eqn | l = {A'}_x
| r = A_x \cos \varphi + A_y \sin \... | Characteristics of Vector in Plane | https://proofwiki.org/wiki/Characteristics_of_Vector_in_Plane | https://proofwiki.org/wiki/Characteristics_of_Vector_in_Plane | [
"Characteristics of Vector in Plane",
"Cartesian Coordinate Systems",
"Vectors"
] | [
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Ordered Pair",
"Definition:Real Number",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Terminal Point of Vector",
"Definition:Position Vector",
"Definition:Linear Transformation",
"Definiti... | [
"Definition:Terminal Point of Vector",
"Definition:Position Vector",
"Rotation of Cartesian Axes around Vector",
"Definition:Position Vector",
"Rotation of Cartesian Axes around Vector"
] |
proofwiki-20725 | Regular Space with Sigma-Locally Finite Basis is Normal | Let $T = \struct {S, \tau}$ be a regular space.
Let there exist a $\sigma$-locally finite basis for $T$.
Then:
:$T$ is a normal space | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
{{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 2}}
From $T_3$ Space with Sigma-Locally Finite Basis is $T_4$ Space:
:$T$ is a $T_4$ space
Hence $T$ is a normal space by definition.
{{qed}}
Category:Re... | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular space]].
Let there exist a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]] for $T$.
Then:
:$T$ is a [[Definition:Normal Space|normal space]] | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
{{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 2}}
From [[T3 Space with Sigma-Locally Finite Basis is T4 Space|$T_3$ Space with Sigma-Locally Finite Basis is $T_4$ Space]]:
:$T$ is a [[Definition:T4... | Regular Space with Sigma-Locally Finite Basis is Normal | https://proofwiki.org/wiki/Regular_Space_with_Sigma-Locally_Finite_Basis_is_Normal | https://proofwiki.org/wiki/Regular_Space_with_Sigma-Locally_Finite_Basis_is_Normal | [
"Regular Spaces",
"Sigma-Locally Finite Bases",
"Normal Spaces"
] | [
"Definition:Regular Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Normal Space"
] | [
"T3 Space with Sigma-Locally Finite Basis is T4 Space",
"Definition:T4 Space",
"Definition:Normal Space",
"Category:Regular Spaces",
"Category:Sigma-Locally Finite Bases",
"Category:Normal Spaces"
] |
proofwiki-20726 | Zero Vector is Orthogonal to All Vectors | Let $\mathbf V$ be a vector space of $n$ dimensions.
Let $\bszero \in \mathbf V$ be the zero vector.
Let $\mathbf a \in \mathbf V$ be an arbitrary vector in $\mathbf V$
Then $\bszero$ is orthogonal to $\mathbf a$. | By definition, $\mathbf a$ and $\mathbf b$ are orthogonal {{iff}} their dot product is zero:
:$\mathbf a \cdot \mathbf b = 0$
We have:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = \sum_{k \mathop = 1}^n a_k \mathbf e_k
}}
{{eqn | l = \mathbf b
| r = \sum_{k \mathop = 1}^n b_k \mathbf e_k
}}
{{end-eqn}}
where $... | Let $\mathbf V$ be a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $\bszero \in \mathbf V$ be the [[Definition:Zero Vector|zero vector]].
Let $\mathbf a \in \mathbf V$ be an arbitrary [[Definition:Vector|vector]] in $\mathbf V$
Then $\bszero$ is [[Definitio... | By definition, $\mathbf a$ and $\mathbf b$ are [[Definition:Orthogonal Vectors|orthogonal]] {{iff}} their [[Definition:Dot Product|dot product]] is [[Definition:Zero (Number)|zero]]:
:$\mathbf a \cdot \mathbf b = 0$
We have:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = \sum_{k \mathop = 1}^n a_k \mathbf e_k
}}
{... | Zero Vector is Orthogonal to All Vectors | https://proofwiki.org/wiki/Zero_Vector_is_Orthogonal_to_All_Vectors | https://proofwiki.org/wiki/Zero_Vector_is_Orthogonal_to_All_Vectors | [
"Zero Vectors",
"Orthogonality (Linear Algebra)"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Zero Vector",
"Definition:Vector",
"Definition:Orthogonal (Linear Algebra)/Real Vector Space"
] | [
"Definition:Orthogonal (Linear Algebra)/Real Vector Space",
"Definition:Dot Product",
"Definition:Zero (Number)",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Dot Product/General Context",
"Definition:Zero Vector"
] |
proofwiki-20727 | Standard Ordered Basis Vectors are Orthogonal | Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over a field $\mathbb F$, as defined by the vector space axioms.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.
Let $\mathbf e_i$ and $\mathbf e_j$ be elements of $\tuple {\mathbf e_1, \mathbf e... | By definition of standard ordered basis:
:$\mathbf e_i$ is a vector whose $i$th component is $1_{\mathbb F}$ with all other components $0_{\mathbb F}$.
Hence:
:$\mathbf e_i \cdot \mathbf e_j = \delta_{i j}$
where $\delta_{i j}$ denotes the Kronecker delta.
That is:
:$i \ne j \implies \mathbf e_i \cdot \mathbf e_j = 0$
... | Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $\mathbb F$, as defined by the [[Axiom:Vector Space Axioms|vector space axioms]].
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the [[Definition:Standard ... | By definition of [[Definition:Standard Ordered Basis|standard ordered basis]]:
:$\mathbf e_i$ is a [[Definition:Vector (Linear Algebra)|vector]] whose $i$th [[Definition:Component of Vector|component]] is $1_{\mathbb F}$ with all other [[Definition:Component of Vector|components]] $0_{\mathbb F}$.
Hence:
:$\mathbf e_... | Standard Ordered Basis Vectors are Orthogonal | https://proofwiki.org/wiki/Standard_Ordered_Basis_Vectors_are_Orthogonal | https://proofwiki.org/wiki/Standard_Ordered_Basis_Vectors_are_Orthogonal | [
"Orthogonality (Linear Algebra)",
"Standard Ordered Bases"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Axiom:Vector Space Axioms",
"Definition:Standard Ordered Basis",
"Definition:Element",
"Definition:Orthogonal (Linear Algebra)/Real Vector Space"
] | [
"Definition:Standard Ordered Basis",
"Definition:Vector/Linear Algebra",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component",
"Definition:Kronecker Delta",
"Definition:Orthogonal (Linear Algebra)/Real Vector Space"
] |
proofwiki-20728 | Open Cover with Closed Locally Finite Refinement is Even Cover | Let $T = \struct {X, \tau}$ be a topological Space.
Let $\UU$ be an open cover of $T$ with a closed locally finite refinement.
Then:
:$\UU$ is an even cover. | Let $\AA$ be a closed locally finite refinement of $\UU$.
By definition of refinement:
:$\forall A \in \AA : \exists U \in \UU : A \subseteq U$
For each $A \in \AA$, let $U_A \in \UU$ such that $A \subseteq U_A$.
Let $T \times T = \struct {X \times X, \tau_{X \times X} }$ denote the product space of $T$ with itself.
F... | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological Space]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$ with a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite]] [[Definition:Refinement of Cover|refinement]].
Then:
:$\UU$ is an [[Definition:Even Cover|e... | Let $\AA$ be a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite]] [[Definition:Refinement of Cover|refinement]] of $\UU$.
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\forall A \in \AA : \exists U \in \UU : A \subseteq U$
For each $A \in \AA$, let $U_A \in \UU$ such that $... | Open Cover with Closed Locally Finite Refinement is Even Cover | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover | [
"Open Cover with Closed Locally Finite Refinement is Even Cover",
"Open Covers",
"Even Covers"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Even Cover"
] | [
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Refinement of Cover",
"Definition:Product Space",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Image (Set Theory)/Relation/Element"
] |
proofwiki-20729 | Equivalence of Definitions of Orthogonal Matrix | {{TFAE|def = Orthogonal Matrix}}
Let $\mathbf Q$ be a nonsingular square matrix over a field $\GF$. | === Definition $(1)$ is equivalent to Definition $(2)$ ===
{{begin-eqn}}
{{eqn | l = \mathbf Q^{-1}
| r = \mathbf Q^\intercal
| c = {{Defof|Orthogonal Matrix|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \mathbf Q^{-1} \mathbf Q
| r = \mathbf Q^\intercal \mathbf Q
| c =
}}
{{eqn | ll= ... | {{TFAE|def = Orthogonal Matrix}}
Let $\mathbf Q$ be a [[Definition:Nonsingular Matrix|nonsingular]] [[Definition:Square Matrix|square matrix]] over a [[Definition:Field (Abstract Algebra)|field]] $\GF$. | === Definition $(1)$ is [[Definition:Logical Equivalence|equivalent]] to Definition $(2)$ ===
{{begin-eqn}}
{{eqn | l = \mathbf Q^{-1}
| r = \mathbf Q^\intercal
| c = {{Defof|Orthogonal Matrix|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \mathbf Q^{-1} \mathbf Q
| r = \mathbf Q^\intercal \m... | Equivalence of Definitions of Orthogonal Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Orthogonal_Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Orthogonal_Matrix | [
"Orthogonal Matrices"
] | [
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Logical Equivalence",
"Definition:Logical Equivalence"
] |
proofwiki-20730 | Set is Neighborhood of Subset iff Neighborhood of all Points of Subset | Let $T = \struct {S, \tau}$ be a topological space.
Let $N \subseteq S$ be a subset of $T$.
Let $A \subseteq N$ be a subset of $T$.
Then:
:$N$ is a neighborhood of $A$ in $T$
{{iff}}:
:$N$ is a neighborhood of all points in $A$ | === Necessary Condition ===
Let $N$ be a neighborhood of $A$ in $T$.
By definition of neighborhood of set:
:$\exists U \in \tau : A \subseteq U \subseteq N$
Let $z \in A$.
By definition of subset:
:$z \in U$
From Set is Open iff Neighborhood of all its Points:
:$U$ is a neighborhood of $z$
From Superset of Neighborhood... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $N \subseteq S$ be a [[Definition:Subset|subset]] of $T$.
Let $A \subseteq N$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$N$ is a [[Definition:Neighborhood of Set|neighborhood]] of $A$ in $T$
{{iff}}:
:$N$ is a [[Definiti... | === Necessary Condition ===
Let $N$ be a [[Definition:Neighborhood of Set|neighborhood]] of $A$ in $T$.
By definition of [[Definition:Neighborhood of Set|neighborhood of set]]:
:$\exists U \in \tau : A \subseteq U \subseteq N$
Let $z \in A$.
By definition of [[Definition:Subset|subset]]:
:$z \in U$
From [[Set is ... | Set is Neighborhood of Subset iff Neighborhood of all Points of Subset | https://proofwiki.org/wiki/Set_is_Neighborhood_of_Subset_iff_Neighborhood_of_all_Points_of_Subset | https://proofwiki.org/wiki/Set_is_Neighborhood_of_Subset_iff_Neighborhood_of_all_Points_of_Subset | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Subset",
"Definition:Neighborhood (Topology)/Set",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Set",
"Definition:Neighborhood (Topology)/Set",
"Definition:Subset",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Superset of Neighborhood in Topological Space is Neighborhood",
"Definition:Neighborhood (Topology)/Poin... |
proofwiki-20731 | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 4 | :$\forall A \in \AA : V_A$ is an open neighborhood of the diagonal $\Delta_X$ in $T \times T$ | Let $A \in \AA$.
By definition of closed set:
:$X \setminus A$ is open in $T$
By definition of product topology:
:$U_A \times U_A, \paren {X \setminus A} \times \paren {X \setminus A}$ are open in $T \times T$
By {{Open-set-axiom|1}}:
:$V_A = \paren {U_A \times U_A} \cup \paren {\paren {X \setminus A} \times \paren {X ... | :$\forall A \in \AA : V_A$ is an [[Definition:Open Neighborhood|open neighborhood]] of the [[Definition:Diagonal Relation|diagonal]] $\Delta_X$ in $T \times T$ | Let $A \in \AA$.
By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$X \setminus A$ is [[Definition:Open Set (Topology)|open]] in $T$
By definition of [[Definition:Product Topology|product topology]]:
:$U_A \times U_A, \paren {X \setminus A} \times \paren {X \setminus A}$ are [[Definition:Open Set (T... | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 4 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_4 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_4 | [
"Open Cover with Closed Locally Finite Refinement is Even Cover"
] | [
"Definition:Open Neighborhood",
"Definition:Diagonal Relation"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Product Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Cartesian Product",
"Definition:Set Union",
"Definition:Diagonal Relation",... |
proofwiki-20732 | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 3 | :$\forall A \in \AA, x \in A : \map {V_A} x = U_A$ | We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \AA, x \in A
| l = \map {V_A} x
| r = \set {y \in X : \tuple{x, y} \in V_A}
| c = Definition of $\map {V_A} x$
}}
{{eqn | r = \set {y \in X : \tuple {x, y} \in \paren {U_A \times U_A} \cup \paren {\paren {X \setminus A} \times \paren {X \setminus A} } }... | :$\forall A \in \AA, x \in A : \map {V_A} x = U_A$ | We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \AA, x \in A
| l = \map {V_A} x
| r = \set {y \in X : \tuple{x, y} \in V_A}
| c = Definition of $\map {V_A} x$
}}
{{eqn | r = \set {y \in X : \tuple {x, y} \in \paren {U_A \times U_A} \cup \paren {\paren {X \setminus A} \times \paren {X \setminus A} } }... | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 3 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_3 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_3 | [
"Open Cover with Closed Locally Finite Refinement is Even Cover"
] | [] | [
"Category:Open Cover with Closed Locally Finite Refinement is Even Cover"
] |
proofwiki-20733 | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 1 | :$\set {\map V x : x \in X}$ is a refinement of $\UU$. | Let $x \in X$.
By definition of refinement:
:$\AA$ is a cover of $X$
By definition of cover:
:$\exists A \in \AA : x \in A$ | :$\set {\map V x : x \in X}$ is a [[Definition:Refinement of Cover|refinement]] of $\UU$. | Let $x \in X$.
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\AA$ is a [[Definition:Cover of Set|cover]] of $X$
By definition of [[Definition:Cover of Set|cover]]:
:$\exists A \in \AA : x \in A$ | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 1 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_1 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_1 | [
"Open Cover with Closed Locally Finite Refinement is Even Cover"
] | [
"Definition:Refinement of Cover"
] | [
"Definition:Refinement of Cover",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:Refinement of Cover"
] |
proofwiki-20734 | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 2 | :$V$ is a neighborhood of the diagonal $\Delta_X$ in $T \times T$. | Let $x \in X$.
By definition of locally finite:
:$\exists W \in \tau : x \in W : \set {A \in \AA : W \cap A \ne \O}$ is finite.
Let:
:$A \in \AA : W \cap A = \O$
From Subset of Set Difference iff Disjoint Set:
:$W \subseteq X \setminus A$
From Cartesian Product of Subsets:
:$W \times W \subseteq \paren {X \setminus A} ... | :$V$ is a [[Definition:Neighborhood of Set|neighborhood]] of the [[Definition:Diagonal Relation|diagonal]] $\Delta_X$ in $T \times T$. | Let $x \in X$.
By definition of [[Definition:Locally Finite Set of Subsets|locally finite]]:
:$\exists W \in \tau : x \in W : \set {A \in \AA : W \cap A \ne \O}$ is [[Definition:Finite Set|finite]].
Let:
:$A \in \AA : W \cap A = \O$
From [[Subset of Set Difference iff Disjoint Set]]:
:$W \subseteq X \setminus A$
... | Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 2 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_2 | https://proofwiki.org/wiki/Open_Cover_with_Closed_Locally_Finite_Refinement_is_Even_Cover/Lemma_2 | [
"Open Cover with Closed Locally Finite Refinement is Even Cover"
] | [
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation"
] | [
"Definition:Locally Finite Set of Subsets",
"Definition:Finite Set",
"Subset of Set Difference iff Disjoint Set",
"Cartesian Product of Subsets",
"Set is Subset of Intersection of Supersets",
"Intersection with Subset is Subset",
"Definition:Product Topology",
"Definition:Open Set/Topology",
"Defini... |
proofwiki-20735 | Angle between Vector Quantities in terms of Direction Cosines | Let $\mathbf a$ and $\mathbf b$ be vector quantities embedded in Cartesian $3$-space
Let $\theta$ be the angle between $\mathbf a$ and $\mathbf b$.
Then:
:$\cos \theta = \lambda_a \lambda_b + \mu_a \mu_b + \nu_a \nu_b$
where $\lambda_a$, $\mu_a$ and $\nu_a$ are the direction cosines of $\mathbf a$ with respect to the $... | Let $\mathbf r$ be an arbitrary vector quantity embedded in a Cartesian $3$-space.
From Components of Vector in terms of Direction Cosines:
{{begin-eqn}}
{{eqn | l = x
| r = r \lambda_r
}}
{{eqn | l = y
| r = r \mu_r
}}
{{eqn | l = z
| r = r \nu_r
}}
{{end-eqn}}
where:
:$x$, $y$ and $z$ denote the com... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]] embedded in [[Definition:Cartesian 3-Space|Cartesian $3$-space]]
Let $\theta$ be the [[Definition:Angle Between Vectors|angle]] between $\mathbf a$ and $\mathbf b$.
Then:
:$\cos \theta = \lambda_a \lambda_b + \mu_a \mu_b + \nu_a \nu_b... | Let $\mathbf r$ be an arbitrary [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
From [[Components of Vector in terms of Direction Cosines]]:
{{begin-eqn}}
{{eqn | l = x
| r = r \lambda_r
}}
{{eqn | l = y
| r = r \mu_r
}}
{{eqn | l = z
... | Angle between Vector Quantities in terms of Direction Cosines | https://proofwiki.org/wiki/Angle_between_Vector_Quantities_in_terms_of_Direction_Cosines | https://proofwiki.org/wiki/Angle_between_Vector_Quantities_in_terms_of_Direction_Cosines | [
"Direction Cosines"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Angle between Vectors",
"Definition:Direction Cosines",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Components of Vector in terms of Direction Cosines",
"Definition:Vector Quantity/Component",
"Definition:Direction",
"Definition:Magnitude"
] |
proofwiki-20736 | Hex Theorem/Corollary | Every game of Hex has exactly one winner. | {{AimForCont}} there is a sequence of moves:
:$P_1, Q_1, P_2, Q_2, \dotsc$
such that neither player wins.
By the Pigeonhole Principle, every tile is marked after $n^2$ moves.
But, by the Hex Theorem, that board has exactly one winner.
This contradicts the assertion that neither player wins.
Thus, by Proof by Contradict... | Every [[Definition:Game|game]] of [[Definition:Hex (Game)|Hex]] has exactly one [[Definition:Hex Winner|winner]]. | {{AimForCont}} there is a [[Definition:Sequence|sequence]] of [[Definition:Hex Move|moves]]:
:$P_1, Q_1, P_2, Q_2, \dotsc$
such that neither [[Definition:Hex Player|player]] [[Definition:Hex Winner|wins]].
By the [[Pigeonhole Principle]], every [[Definition:Tile of Hex Board|tile]] is marked after $n^2$ moves.
But, b... | Hex Theorem/Corollary | https://proofwiki.org/wiki/Hex_Theorem/Corollary | https://proofwiki.org/wiki/Hex_Theorem/Corollary | [
"Hex Theorem"
] | [
"Definition:Game",
"Definition:Hex (Game)",
"Definition:Hex (Game)/Winning"
] | [
"Definition:Sequence",
"Definition:Hex (Game)/Moves",
"Definition:Hex (Game)/Player",
"Definition:Hex (Game)/Winning",
"Dirichlet's Box Principle/Corollary",
"Definition:Hex (Game)/Board/Tile",
"Hex Theorem",
"Definition:Hex (Game)/Board",
"Definition:Hex (Game)/Winning",
"Definition:Contradiction... |
proofwiki-20737 | Lévy's Inversion Formula | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $ F_X$ be the distribution function of $X$.
Let $\phi : \R \to \C$ be the characteristic function of $X$.
Then for all $a < b$ such that:
:$\map \Pr {X \in \set {a,b} } = 0$
we ha... | In fact, the first equality holds for all $a < b$, as:
{{begin-eqn}}
{{eqn | l = \map {F_X} b - \map {F_X} a
| r = \map \Pr {X \le b} - \map \Pr {X \le a}
| c = {{Defof|Cumulative Distribution Function|$F_X$}}
}}
{{eqn | r = \map \Pr {\set {X \le b} \setminus \set {X \le a} }
| c = Measure of Set Diff... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $ F_X$ be the [[Definition:Cumulative Distribution Function|distribution function]] of $X$.
Let $\ph... | In fact, the first equality holds for all $a < b$, as:
{{begin-eqn}}
{{eqn | l = \map {F_X} b - \map {F_X} a
| r = \map \Pr {X \le b} - \map \Pr {X \le a}
| c = {{Defof|Cumulative Distribution Function|$F_X$}}
}}
{{eqn | r = \map \Pr {\set {X \le b} \setminus \set {X \le a} }
| c = [[Measure of Set Di... | Lévy's Inversion Formula | https://proofwiki.org/wiki/Lévy's_Inversion_Formula | https://proofwiki.org/wiki/Lévy's_Inversion_Formula | [
"Lévy's Inversion Formula",
"Probability Theory"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Cumulative Distribution Function",
"Definition:Characteristic Function of Random Variable"
] | [
"Measure of Set Difference with Subset",
"Definition:Probability Distribution/Real-Valued Random Variable",
"Definition:Lebesgue Measure",
"Definition:Restriction of Measure to Trace Sigma-Algebra of Measurable Set",
"Definition:Product Measure",
"Definition:Essentially Bounded Function",
"Bounds for Co... |
proofwiki-20738 | Characterization of Paracompactness in T3 Space/Lemma 1 | :$\VV$ is an open cover of $T$ | Let $x \in S$.
By definition of open cover:
:$\exists U \in \UU : x \in U$
From Characterization of T3 Space:
:$\exists V \in \tau : x \in V : V^- \subseteq U$
Hence:
:$V \in \VV$
Since $x$ was arbitrary, $\VV$ is an open cover by definition.
{{qed}}
Category:Characterization of Paracompactness in T3 Space
1t9o2hrsoy6m... | :$\VV$ is an [[Definition:Open Cover|open cover]] of $T$ | Let $x \in S$.
By definition of [[Definition:Open Cover|open cover]]:
:$\exists U \in \UU : x \in U$
From [[Characterization of T3 Space]]:
:$\exists V \in \tau : x \in V : V^- \subseteq U$
Hence:
:$V \in \VV$
Since $x$ was arbitrary, $\VV$ is an [[Definition:Open Cover|open cover]] by definition.
{{qed}}
[[Ca... | Characterization of Paracompactness in T3 Space/Lemma 1 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_1 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_1 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Open Cover"
] | [
"Definition:Open Cover",
"Characterization of T3 Space",
"Definition:Open Cover",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20739 | Characterization of Paracompactness in T3 Space/Lemma 2 | :$\BB$ is a cover of $T$ consisting of closed sets | Let $x \in X$.
By definition of refinement:
:$\AA$ is a cover of $X$
By definition of cover of set:
:$\exists A \in \AA : x \in A$
From Set is Subset of its Topological Closure:
:$A \subseteq A^-$
By definition of subset:
:$x \in A^-$
By definition of $\BB$:
:$A^- \in \BB$
Since $x$ was arbitrary, $\BB$ is a cover of $... | :$\BB$ is a [[Definition:Cover of Set|cover]] of $T$ consisting of [[Definition:Closed Set (Topology)|closed sets]] | Let $x \in X$.
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\AA$ is a [[Definition:Cover of Set|cover]] of $X$
By definition of [[Definition:Cover of Set|cover of set]]:
:$\exists A \in \AA : x \in A$
From [[Set is Subset of its Topological Closure]]:
:$A \subseteq A^-$
By definition of [[D... | Characterization of Paracompactness in T3 Space/Lemma 2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_2 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Cover of Set",
"Definition:Closed Set/Topology"
] | [
"Definition:Refinement of Cover",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Set is Subset of its Topological Closure",
"Definition:Subset",
"Definition:Cover of Set",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Category:Characterization of Paracompactness in T3 S... |
proofwiki-20740 | Characterization of Paracompactness in T3 Space/Lemma 3 | :$\BB$ is a refinement of $\UU$ | ==== Lemma 2 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 2}}{{qed|lemma}}
Let $B \in \BB$.
By definition of $\BB$:
:$\exists A \in \AA : A^- = B$
By definition of refinement:
:$\exists V \in \VV : A \subseteq V$
From Set Closure Preserves Set Inclusion:
:$B = A^- \subseteq V^-$
By definition of $\VV$:... | :$\BB$ is a [[Definition:Refinement of Cover|refinement]] of $\UU$ | ==== [[Characterization of Paracompactness in T3 Space/Lemma 2|Lemma 2]] ====
{{:Characterization of Paracompactness in T3 Space/Lemma 2}}{{qed|lemma}}
Let $B \in \BB$.
By definition of $\BB$:
:$\exists A \in \AA : A^- = B$
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\exists V \in \VV : A \su... | Characterization of Paracompactness in T3 Space/Lemma 3 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_3 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_3 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Refinement of Cover"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 2",
"Definition:Refinement of Cover",
"Set Closure Preserves Set Inclusion",
"Subset Relation is Transitive",
"Definition:Refinement of Cover",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20741 | Cross Product of Parallel Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities which are parallel.
Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \mathbf 0$
where $\mathbf 0$ denotes the zero vector. | By definition of cross product:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
:$\hat {\mathbf n}$ is the unit vector perpendicular to bot... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]] which are [[Definition:Parallel Lines|parallel]].
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|cross product]] of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \mathbf 0$
where $\... | By definition of [[Definition:Vector Cross Product/Definition 2|cross product]]:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the [[Definition:Vector Length|length]] of $\mathbf a$
:$\theta$ denotes the [[Definition:Angle Between Vectors|angle]] from $\mat... | Cross Product of Parallel Vectors | https://proofwiki.org/wiki/Cross_Product_of_Parallel_Vectors | https://proofwiki.org/wiki/Cross_Product_of_Parallel_Vectors | [
"Vector Cross Product"
] | [
"Definition:Vector Quantity",
"Definition:Parallel (Geometry)/Lines",
"Definition:Vector Cross Product",
"Definition:Zero Vector"
] | [
"Definition:Vector Cross Product/Definition 2",
"Definition:Vector Length",
"Definition:Angle between Vectors",
"Definition:Axis/Positive Direction",
"Definition:Unit Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Right-Hand Rule/Cross Product",
"Definition:Parallel (Geometry)/Lines",
... |
proofwiki-20742 | Rotating Indices Property of Vector Triple Product | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Euclidean $3$-space.
Then:
:$\mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b} = 0$
where $\mathbf a \times \paren {\mathbf b \times \mathbf c}$ d... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b}
| c =
}}
{{eqn | r = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf a \cdot \mathbf b} \mathbf ... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Vector Quantity|vectors]] in a [[Definition:Real Euclidean Space|Euclidean $3$-space]].
Then:
:$\mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b}... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b}
| c =
}}
{{eqn | r = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf a \cdot \mathbf b} \mathbf ... | Rotating Indices Property of Vector Triple Product | https://proofwiki.org/wiki/Rotating_Indices_Property_of_Vector_Triple_Product | https://proofwiki.org/wiki/Rotating_Indices_Property_of_Vector_Triple_Product | [
"Vector Triple Product"
] | [
"Definition:Vector Quantity",
"Definition:Euclidean Space/Real",
"Definition:Vector Triple Product"
] | [
"Lagrange's Formula",
"Definition:Dot Product",
"Dot Product Operator is Commutative"
] |
proofwiki-20743 | Square of Vector Cross Product | Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions.
Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.
Then:
:$\paren {\mathbf a \times \mathbf b}^2 = \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b}^2$
where:
:$\paren {\... | Let $\theta$ denote the angle between $\mathbf a$ and $\mathbf b$
{{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b}^2
| r = \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf a \times \mathbf b}
| c = {{Defof|Square of Vector Quantity}}
}}
{{eqn | r = \paren {\norm {\mathbf a} \norm {\mat... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] $\mathbf V$ of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|vector cross product]] of $\mathbf a$ wit... | Let $\theta$ denote the [[Definition:Angle Between Vectors|angle]] between $\mathbf a$ and $\mathbf b$
{{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b}^2
| r = \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf a \times \mathbf b}
| c = {{Defof|Square of Vector Quantity}}
}}
{{eqn | r =... | Square of Vector Cross Product/Proof 2 | https://proofwiki.org/wiki/Square_of_Vector_Cross_Product | https://proofwiki.org/wiki/Square_of_Vector_Cross_Product/Proof_2 | [
"Square of Vector Cross Product",
"Vector Cross Product",
"Dot Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product",
"Definition:Square of Vector Quantity",
"Definition:Dot Product"
] | [
"Definition:Angle between Vectors",
"Definition:Unit Vector",
"Dot Product of Vector with Itself",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-20744 | Order of Simple Group divides Factorial of Index of Subgroup | Let $G$ be a simple group, and let $H$ be a subgroup of $G$ with index $n$. Then $|G|$ divides $n!$. | Let $G$ act on the cosets of $H$, $G/H$ by the left regular action.
By Group Action defines Permutation Representation, this defines a homomorphism $\tilde \phi: G \to \map \Gamma X$.
By Kernel is Normal Subgroup of Domain, $\ker \tilde \phi$ is a normal subgroup of $G$.
Since $G$ is simple, $\ker \tilde \phi = \{e\}$ ... | Let $G$ be a simple group, and let $H$ be a subgroup of $G$ with index $n$. Then $|G|$ divides $n!$. | Let $G$ act on the cosets of $H$, $G/H$ by the left regular action.
By [[Group Action defines Permutation Representation]], this defines a homomorphism $\tilde \phi: G \to \map \Gamma X$.
By [[Kernel is Normal Subgroup of Domain]], $\ker \tilde \phi$ is a normal subgroup of $G$.
Since $G$ is simple, $\ker \tilde \ph... | Order of Simple Group divides Factorial of Index of Subgroup | https://proofwiki.org/wiki/Order_of_Simple_Group_divides_Factorial_of_Index_of_Subgroup | https://proofwiki.org/wiki/Order_of_Simple_Group_divides_Factorial_of_Index_of_Subgroup | [] | [] | [
"Group Action defines Permutation Representation",
"Kernel is Normal Subgroup of Domain",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-20745 | Number of Sylow p-Subgroups is Index of Normalizer of Sylow p-Subgroup | Let $P$ be a Sylow p-subgroup of a group $G$, $N_G (S)$ its normalizer, and $n_p$ be the number of Sylow p-subgroups in $G$.
Then $[ G : N_G (S) ] = n_p$. | By Number of Distinct Conjugate Subsets is Index of Normalizer, $[ G : N_G (S) ] = n_p$.
{{qed}}
dngs99iml7th8u6v95kmmnlthoaatmo | Let $P$ be a Sylow p-subgroup of a group $G$, $N_G (S)$ its [[Definition:Normalizer|normalizer]], and $n_p$ be the number of Sylow p-subgroups in $G$.
Then $[ G : N_G (S) ] = n_p$. | By [[Number of Distinct Conjugate Subsets is Index of Normalizer]], $[ G : N_G (S) ] = n_p$.
{{qed}}
dngs99iml7th8u6v95kmmnlthoaatmo | Number of Sylow p-Subgroups is Index of Normalizer of Sylow p-Subgroup | https://proofwiki.org/wiki/Number_of_Sylow_p-Subgroups_is_Index_of_Normalizer_of_Sylow_p-Subgroup | https://proofwiki.org/wiki/Number_of_Sylow_p-Subgroups_is_Index_of_Normalizer_of_Sylow_p-Subgroup | [] | [
"Definition:Normalizer"
] | [
"Number of Distinct Conjugate Subsets is Index of Normalizer"
] |
proofwiki-20746 | Group of Order 36 is not Simple | Let $G$ be of order $36$.
Then $G$ is not simple. | {{AimForCont}} $G$ is simple.
We have that:
:$36 = 2^2 \times 3^2$
Let $n_3$ denote the number of Sylow $3$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_3 \equiv 1 \pmod 3$
and from the Fifth Sylow Theorem:
:$n_3 \divides 4$
where $\divides$ denotes divisibility.
Hence $n_3$ must be either $1$ or $4$.
Let $P$ b... | Let $G$ be of [[Definition:Order of Group|order]] $36$.
Then $G$ is not [[Definition:Simple Group|simple]]. | {{AimForCont}} $G$ is [[Definition:Simple Group|simple]].
We have that:
:$36 = 2^2 \times 3^2$
Let $n_3$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $3$-subgroups]] of $G$.
From the [[Fourth Sylow Theorem]]:
:$n_3 \equiv 1 \pmod 3$
and from the [[Fifth Sylow Theorem]]:
:$n_3 \divides 4$
where $\divides... | Group of Order 36 is not Simple | https://proofwiki.org/wiki/Group_of_Order_36_is_not_Simple | https://proofwiki.org/wiki/Group_of_Order_36_is_not_Simple | [
"Groups of Order 36"
] | [
"Definition:Order of Structure",
"Definition:Simple Group"
] | [
"Definition:Simple Group",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sylow p-Subgroup",
"Number of Sylow p-Subgroups is Index of Normalizer of Sylow p-Subgroup",
"Definition:Normalizer",
"Definition:Index of Subg... |
proofwiki-20747 | Group of Order 48 is not Simple | Let $G$ be of order $48$.
Then $G$ is not simple. | {{AimForCont}} $G$ is simple.
We have that:
:$48 = 2^4 \times 3$
Let $n_2$ denote the number of Sylow $2$-subgroups of $G$.
From Sylow $2$-Subgroups in Group of Order $48$, $n_2$ is either $1$ or $3$.
Let $P$ be a Sylow $2$-subgroup of $G$.
Let $n_2 = 3$.
By Number of Sylow p-Subgroups is Index of Normalizer of Sylow p... | Let $G$ be of [[Definition:Order of Group|order]] $48$.
Then $G$ is not [[Definition:Simple Group|simple]]. | {{AimForCont}} $G$ is [[Definition:Simple Group|simple]].
We have that:
:$48 = 2^4 \times 3$
Let $n_2$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $2$-subgroups]] of $G$.
From [[Sylow Theorems/Examples/Sylow 2-Subgroups in Group of Order 48|Sylow $2$-Subgroups in Group of Order $48$]], $n_2$ is either $... | Group of Order 48 is not Simple | https://proofwiki.org/wiki/Group_of_Order_48_is_not_Simple | https://proofwiki.org/wiki/Group_of_Order_48_is_not_Simple | [
"Groups of Order 48"
] | [
"Definition:Order of Structure",
"Definition:Simple Group"
] | [
"Definition:Simple Group",
"Definition:Sylow p-Subgroup",
"Sylow Theorems/Examples/Sylow 2-Subgroups in Group of Order 48",
"Definition:Sylow p-Subgroup",
"Number of Sylow p-Subgroups is Index of Normalizer of Sylow p-Subgroup",
"Definition:Normalizer",
"Definition:Index of Subgroup",
"Order of Simple... |
proofwiki-20748 | Ordering of Natural Numbers is Provable | Let $x, y \in \N$.
Suppose $x < y$.
Let $\sqbrk a$ denote the unary representation of $a \in \N$.
Then $\sqbrk x < \sqbrk y$ is a theorem of minimal arithmetic. | Fix $x$, and let $z > 0$ be arbitrary.
Proceed by induction on $z$. | Let $x, y \in \N$.
Suppose $x < y$.
Let $\sqbrk a$ denote the [[Unary Representation of Natural Number|unary representation]] of $a \in \N$.
Then $\sqbrk x < \sqbrk y$ is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arithmetic|minimal arithmetic]]. | Fix $x$, and let $z > 0$ be arbitrary.
Proceed by [[Definition:Mathematical Induction|induction]] on $z$. | Ordering of Natural Numbers is Provable | https://proofwiki.org/wiki/Ordering_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Ordering_of_Natural_Numbers_is_Provable | [
"Ordering on Natural Numbers",
"Proofs by Induction"
] | [
"Unary Representation of Natural Number",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-20749 | Sigmoid Function is Strictly Increasing | The real sigmoid function $\map S x$ is strictly increasing.
{{expand|Add a proof based on the derivative being everywhere positive}} | By Cumulative Distribution Function of Logistic Distribution, $S$ is the cumulative distribution function of a logistic distribution, with $\mu = 0$ and $s = 1$.
By Cumulative Distribution Function is Increasing, $S$ is an increasing real function.
{{AimForCont}}, suppose that $S$ is not strictly increasing.
Then there... | The [[Definition:Sigmoid Function|real sigmoid function]] $\map S x$ is [[Definition:Strictly Increasing Real Function|strictly increasing]].
{{expand|Add a proof based on the derivative being everywhere positive}} | By [[Cumulative Distribution Function of Logistic Distribution]], $S$ is the [[Definition:Cumulative Distribution Function|cumulative distribution function]] of a [[Definition:Logistic Distribution|logistic distribution]], with $\mu = 0$ and $s = 1$.
By [[Cumulative Distribution Function is Increasing]], $S$ is an [[D... | Sigmoid Function is Strictly Increasing | https://proofwiki.org/wiki/Sigmoid_Function_is_Strictly_Increasing | https://proofwiki.org/wiki/Sigmoid_Function_is_Strictly_Increasing | [
"Sigmoid Function"
] | [
"Definition:Sigmoid Function",
"Definition:Strictly Increasing/Real Function"
] | [
"Cumulative Distribution Function of Logistic Distribution",
"Definition:Cumulative Distribution Function",
"Definition:Logistic Distribution",
"Cumulative Distribution Function is Increasing",
"Definition:Increasing/Real Function",
"Definition:Strictly Increasing/Real Function",
"Exponential is Strictl... |
proofwiki-20750 | Range of Sigmoid Function | Let $S$ be the real sigmoid function.
Then the range of $S$ is $\openint 0 1$. | By Sigmoid Function is Continuous, $S$ is a continuous real function.
By Image of Real Interval under Continuous Real Function is Real Interval, the range of $S$ is an interval $I$.
By Cumulative Distribution Function of Logistic Distribution, $S$ is the cumulative distribution function of a logistic distribution, with... | Let $S$ be the [[Definition:Sigmoid Function|real sigmoid function]].
Then the [[Definition:Range of Real Function|range]] of $S$ is $\openint 0 1$. | By [[Sigmoid Function is Continuous]], $S$ is a [[Definition:Continuous Real Function|continuous real function]].
By [[Image of Real Interval under Continuous Real Function is Real Interval]], the [[Definition:Range of Real Function|range]] of $S$ is an [[Definition:Real Interval|interval]] $I$.
By [[Cumulative Distr... | Range of Sigmoid Function | https://proofwiki.org/wiki/Range_of_Sigmoid_Function | https://proofwiki.org/wiki/Range_of_Sigmoid_Function | [
"Sigmoid Function"
] | [
"Definition:Sigmoid Function",
"Definition:Real Function/Range"
] | [
"Sigmoid Function is Continuous",
"Definition:Continuous Real Function",
"Image of Real Interval under Continuous Real Function is Real Interval",
"Definition:Real Function/Range",
"Definition:Real Interval",
"Cumulative Distribution Function of Logistic Distribution",
"Definition:Cumulative Distributio... |
proofwiki-20751 | Inequality of Natural Numbers is Provable | Let $x, y \in \N$ be natural numbers.
Suppose $x \ne y$.
Let $\sqbrk a$ denote the unary representation of $a \in \N$.
Then $\sqbrk x \ne \sqbrk y$ is a theorem of minimal arithmetic. | === Lemma ===
{{:Inequality of Natural Numbers is Provable/Lemma}}{{qed|lemma}}
As $x \ne y$, by Ordering on Natural Numbers is Trichotomy, either $x < y$ or $y < x$.
If $y < x$, the result follows from the lemma.
{{qed|lemma}}
If $x < y$, then:
:$\sqbrk y \ne \sqbrk x$
is a theorem by the lemma.
Hence, the following f... | Let $x, y \in \N$ be [[Definition:Natural Number|natural numbers]].
Suppose $x \ne y$.
Let $\sqbrk a$ denote the [[Unary Representation of Natural Number|unary representation]] of $a \in \N$.
Then $\sqbrk x \ne \sqbrk y$ is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arithmetic|minimal ... | === [[Inequality of Natural Numbers is Provable/Lemma|Lemma]] ===
{{:Inequality of Natural Numbers is Provable/Lemma}}{{qed|lemma}}
As $x \ne y$, by [[Ordering on Natural Numbers is Trichotomy]], either $x < y$ or $y < x$.
If $y < x$, the result follows from the [[Inequality of Natural Numbers is Provable/Lemma|lemm... | Inequality of Natural Numbers is Provable | https://proofwiki.org/wiki/Inequality_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Inequality_of_Natural_Numbers_is_Provable | [
"Inequality of Natural Numbers is Provable",
"Natural Numbers",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Unary Representation of Natural Number",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Inequality of Natural Numbers is Provable/Lemma",
"Ordering on Natural Numbers is Trichotomy",
"Inequality of Natural Numbers is Provable/Lemma",
"Definition:Theorem/Formal System",
"Inequality of Natural Numbers is Provable/Lemma",
"Definition:Proof System/Formal Proof",
"Rule of Assumption",
"Equal... |
proofwiki-20752 | Sine Integral Function is Bounded | Let $\Si: \R \to \R$ denote the sine integral function.
Then $\Si$ is bounded. | By Limit at Infinity of Sine Integral Function and its corollary:
{{begin-eqn}}
{{eqn | l = \lim _{x \mathop \to +\infty} \size {\map \Si x}
| r = \lim _{x \mathop \to -\infty} \size {\map \Si x}
}}
{{eqn | r = \dfrac \pi 2
}}
{{end-eqn}}
Thus there is a $M > 0$ such that for all $x \in \R$:
:$\size x > M \implie... | Let $\Si: \R \to \R$ denote the [[Definition:Sine Integral Function|sine integral function]].
Then $\Si$ is [[Definition:Bounded Real-Valued Function |bounded]]. | By [[Limit at Infinity of Sine Integral Function]] and its [[Limit at Infinity of Sine Integral Function/Corollary|corollary]]:
{{begin-eqn}}
{{eqn | l = \lim _{x \mathop \to +\infty} \size {\map \Si x}
| r = \lim _{x \mathop \to -\infty} \size {\map \Si x}
}}
{{eqn | r = \dfrac \pi 2
}}
{{end-eqn}}
Thus there ... | Sine Integral Function is Bounded | https://proofwiki.org/wiki/Sine_Integral_Function_is_Bounded | https://proofwiki.org/wiki/Sine_Integral_Function_is_Bounded | [
"Sine Integral Function"
] | [
"Definition:Sine Integral Function",
"Definition:Bounded Real-Valued Function "
] | [
"Limit at Infinity of Sine Integral Function",
"Limit at Infinity of Sine Integral Function/Corollary",
"Category:Sine Integral Function"
] |
proofwiki-20753 | Negation of Ordering of Natural Numbers is Provable | Let $x, y \in \N$ be natural numbers.
Suppose $x \ge y$.
Let $\sqbrk a$ denote the unary representation of $a \in \N$.
Then:
:$\neg \paren {\sqbrk x < \sqbrk y}$
is a theorem of minimal arithmetic. | Fix $x \in \N$.
Proceed by induction on $y$. | Let $x, y \in \N$ be [[Definition:Natural Number|natural numbers]].
Suppose $x \ge y$.
Let $\sqbrk a$ denote the [[Unary Representation of Natural Number|unary representation]] of $a \in \N$.
Then:
:$\neg \paren {\sqbrk x < \sqbrk y}$
is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arith... | Fix $x \in \N$.
Proceed by [[Definition:Mathematical Induction|induction]] on $y$. | Negation of Ordering of Natural Numbers is Provable | https://proofwiki.org/wiki/Negation_of_Ordering_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Negation_of_Ordering_of_Natural_Numbers_is_Provable | [
"Ordering on Natural Numbers",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Unary Representation of Natural Number",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-20754 | Lévy's Inversion Formula/Integrable Characteristic Function | Let $X$ be a real-valued random variable.
Let $P_X$ be the probability distribution of $X$.
Let $\phi : \R \to \C$ be the characteristic function of $X$.
Suppose that $\phi$ is Lebesgue integrable, i.e.:
:$\ds \int _\R \cmod {\map \phi t} \rd t < + \infty$
Then $P_X$ is absolutely continuous {{WRT}} the Lebesgue measur... | Let $a < b$ be such that:
:$\map {P_X} {\set {a,b} } = 0$
Then:
{{begin-eqn}}
{{eqn | l = \int_a^b \map g x \rd x
| r = \dfrac 1 {2 \pi} \int_a^b \paren {\int_\R \map \phi t e^{- i t x} \rd t} \rd x
}}
{{eqn | r = \dfrac 1 {2 \pi} \int_\R \map \phi t \paren {\int_a^b e^{- i t x} \rd x} \rd t
| c = Fubini's ... | Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]].
Let $P_X$ be the [[Definition:Probability Distribution/Real-Valued Random Variable|probability distribution]] of $X$.
Let $\phi : \R \to \C$ be the [[Definition:Characteristic Function of Random Variable|characteristic function]] of ... | Let $a < b$ be such that:
:$\map {P_X} {\set {a,b} } = 0$
Then:
{{begin-eqn}}
{{eqn | l = \int_a^b \map g x \rd x
| r = \dfrac 1 {2 \pi} \int_a^b \paren {\int_\R \map \phi t e^{- i t x} \rd t} \rd x
}}
{{eqn | r = \dfrac 1 {2 \pi} \int_\R \map \phi t \paren {\int_a^b e^{- i t x} \rd x} \rd t
| c = [[Fubini... | Lévy's Inversion Formula/Integrable Characteristic Function | https://proofwiki.org/wiki/Lévy's_Inversion_Formula/Integrable_Characteristic_Function | https://proofwiki.org/wiki/Lévy's_Inversion_Formula/Integrable_Characteristic_Function | [
"Lévy's Inversion Formula"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Probability Distribution/Real-Valued Random Variable",
"Definition:Characteristic Function of Random Variable",
"Definition:Integrable Function/Lebesgue",
"Definition:Absolute Continuity/Measure",
"Definition:Lebesgue Measure",
"Definition:Radon-Niko... | [
"Fubini's Theorem",
"Lebesgue's Dominated Convergence Theorem",
"Lévy's Inversion Formula",
"Category:Lévy's Inversion Formula"
] |
proofwiki-20755 | Characterization of Paracompactness in T3 Space/Lemma 6 | :$\VV^*$ is an open locally finite cover of $T$ | ==== Lemma 4 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 4}}{{qed|lemma}} | :$\VV^*$ is an [[Definition:Open Locally Finite Set of Subsets|open locally finite]] [[Definition:Open Cover|cover]] of $T$ | ==== [[Characterization of Paracompactness in T3 Space/Lemma 4|Lemma 4]] ====
{{:Characterization of Paracompactness in T3 Space/Lemma 4}}{{qed|lemma}} | Characterization of Paracompactness in T3 Space/Lemma 6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_6 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Open Locally Finite Set of Subsets",
"Definition:Open Cover"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 4"
] |
proofwiki-20756 | Quantifier-Free Formula of Arithmetic is Provable | Let $\phi$ be a sentence in the language of arithmetic.
Suppose $\phi$ contains no quantifiers.
Suppose also that $\N \models \phi$.
:That is, the natural numbers model $\phi$.
Then $\phi$ is a theorem of minimal arithmetic. | By Existence of Negation Normal Form of Statement, $\phi$ is logically equivalent to a WFF $\psi$ such that:
:The only logical connectives are $\set {\neg, \land, \lor}$.
:The connective $\neg$ only appears in front of a predicate symbol.
Proceed by induction on the structure of WFFs in the language of arithmetic. | Let $\phi$ be a [[Definition:Sentence|sentence]] in the [[Definition:Language of Arithmetic|language of arithmetic]].
Suppose $\phi$ contains no [[Definition:Quantifier|quantifiers]].
Suppose also that $\N \models \phi$.
:That is, the [[Definition:Natural Number|natural numbers]] [[Definition:Model (Predicate Logic)|... | By [[Existence of Negation Normal Form of Statement]], $\phi$ is [[Definition:Logical Equivalence|logically equivalent]] to a [[Definition:WFF of Predicate Logic|WFF]] $\psi$ such that:
:The only [[Definition:Logical Connective|logical connectives]] are $\set {\neg, \land, \lor}$.
:The [[Definition:Logical Connective|c... | Quantifier-Free Formula of Arithmetic is Provable | https://proofwiki.org/wiki/Quantifier-Free_Formula_of_Arithmetic_is_Provable | https://proofwiki.org/wiki/Quantifier-Free_Formula_of_Arithmetic_is_Provable | [
"Natural Numbers",
"Proof Theory"
] | [
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Arithmetic",
"Definition:Quantifier",
"Definition:Natural Numbers",
"Definition:Model (Predicate Logic)",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Existence of Negation Normal Form of Statement",
"Definition:Logical Equivalence",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Logical Connective",
"Definition:Logical Connective",
"Definition:Predicate Symbol",
"Definition:Language of Predicate Logic/Formal Grammar",
"Defini... |
proofwiki-20757 | Quantifier-Free Formula of Arithmetic is Provable/Corollary | Let $\phi$ be a sentence in the language of arithmetic.
Suppose $\phi$ contains no quantifiers.
Then exactly one of:
:$\phi$
:$\neg \phi$
is a theorem of minimal arithmetic. | By Law of Excluded Middle, either $\N \models \phi$, or $\N \models \neg \phi$.
By Quantifier-Free Formula of Arithmetic is Provable, whichever one holds is a theorem of minimal arithmetic.
{{qed|lemma}}
Now, suppose that $\N \models \phi$.
Then, $\N \not\models \phi$ by definition of value of formula.
Thus, by the def... | Let $\phi$ be a [[Definition:Sentence|sentence]] in the [[Definition:Language of Arithmetic|language of arithmetic]].
Suppose $\phi$ contains no [[Definition:Quantifier|quantifiers]].
Then exactly one of:
:$\phi$
:$\neg \phi$
is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arithmetic|mini... | By [[Law of Excluded Middle]], either $\N \models \phi$, or $\N \models \neg \phi$.
By [[Quantifier-Free Formula of Arithmetic is Provable]], whichever one holds is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arithmetic|minimal arithmetic]].
{{qed|lemma}}
Now, suppose that $\N \models \... | Quantifier-Free Formula of Arithmetic is Provable/Corollary | https://proofwiki.org/wiki/Quantifier-Free_Formula_of_Arithmetic_is_Provable/Corollary | https://proofwiki.org/wiki/Quantifier-Free_Formula_of_Arithmetic_is_Provable/Corollary | [] | [
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Arithmetic",
"Definition:Quantifier",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Law of Excluded Middle",
"Quantifier-Free Formula of Arithmetic is Provable",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic",
"Definition:Value of Formula under Assignment",
"Definition:Sound Proof System",
"Definition:Theorem/Formal System",
"Definition:Theorem/Formal System"
] |
proofwiki-20758 | Characteristic Function of Random Variable is Well-Defined | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
The characteristic function $\phi: \R \to \C$ of $X$ is well-defined. | Let $t \in \R$.
Recall:
{{begin-eqn}}
{{eqn | l = \map \phi t
| r = \expect {e^{i t X} }
| c = {{Defof|Characteristic Function of Random Variable|Characteristic Function}}
}}
{{eqn | r = \int e^{i t X} \rd \Pr
| c = {{Defof|Expectation/General Definition|Expectation}}
}}
{{end-eqn}}
Thus we need to sh... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
The [[Definition:Characteristic Function of Random Variable|characteristic function]] $\phi: \R \to \C$ ... | Let $t \in \R$.
Recall:
{{begin-eqn}}
{{eqn | l = \map \phi t
| r = \expect {e^{i t X} }
| c = {{Defof|Characteristic Function of Random Variable|Characteristic Function}}
}}
{{eqn | r = \int e^{i t X} \rd \Pr
| c = {{Defof|Expectation/General Definition|Expectation}}
}}
{{end-eqn}}
Thus we need to... | Characteristic Function of Random Variable is Well-Defined | https://proofwiki.org/wiki/Characteristic_Function_of_Random_Variable_is_Well-Defined | https://proofwiki.org/wiki/Characteristic_Function_of_Random_Variable_is_Well-Defined | [
"Probability Theory"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Characteristic Function of Random Variable",
"Definition:Well-Defined"
] | [
"Modulus of Exponential of Imaginary Number is One",
"Characterization of Integrable Functions",
"Category:Probability Theory"
] |
proofwiki-20759 | Existential Quantification of Provable Arithmetic Formula is Provable | Let $\map \phi x$ be a WFF in the language of arithmetic with $1$ free variable.
Let $\sqbrk a$ denote the unary representation of $a$.
Suppose that, for every $n \in \N$ such that $\N \models \map \phi {x \gets \sqbrk n}$:
:$\map \phi {x \gets \sqbrk n}$
is a theorem of minimal arithmetic.
Suppose also that:
:$\N \mod... | By definition of value of formula:
:$\exists x \in \N: \map \phi x$
Choose $x_0 \in \N$ that satisfies $\map \phi {x \gets x_0}$.
By Unary Representation of Natural Number:
:$x_0 = \sqbrk {x_0}$
Thus, by Substitution Property of Equality:
:$\map \phi {x \gets \sqbrk {x_0} }$
But then, by hypothesis:
:$\map \phi {x \get... | Let $\map \phi x$ be a [[Definition:WFF of Predicate Logic|WFF]] in the [[Definition:Language of Arithmetic|language of arithmetic]] with $1$ [[Definition:Free Variable (Predicate Logic)|free variable]].
Let $\sqbrk a$ denote the [[Unary Representation of Natural Number|unary representation]] of $a$.
Suppose that, fo... | By definition of [[Definition:Value of Formula under Assignment|value of formula]]:
:$\exists x \in \N: \map \phi x$
Choose $x_0 \in \N$ that satisfies $\map \phi {x \gets x_0}$.
By [[Unary Representation of Natural Number]]:
:$x_0 = \sqbrk {x_0}$
Thus, by [[Substitution Property of Equality]]:
:$\map \phi {x \gets ... | Existential Quantification of Provable Arithmetic Formula is Provable | https://proofwiki.org/wiki/Existential_Quantification_of_Provable_Arithmetic_Formula_is_Provable | https://proofwiki.org/wiki/Existential_Quantification_of_Provable_Arithmetic_Formula_is_Provable | [
"Natural Numbers",
"Proof Theory"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Language of Arithmetic",
"Definition:Free Variable/Predicate Logic",
"Unary Representation of Natural Number",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic",
"Definition:Theorem/Formal System",
"Definition:Minim... | [
"Definition:Value of Formula under Assignment",
"Unary Representation of Natural Number",
"Substitution Property of Equality",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic",
"Existential Generalisation",
"Category:Natural Numbers",
"Category:Proof Theory"
] |
proofwiki-20760 | Bounded Universal Quantification of Provable Arithmetic Formula is Provable | Let $\map \phi x$ be a WFF in the language of arithmetic with $1$ free variable.
Let $\tau$ be a term in the language of arithmetic with no variables.
Let $\sqbrk a$ denote the unary representation of $a$.
Suppose that, for every $n \in \N$ such that $\N \models \map \phi {x \gets \sqbrk n}$:
:$\map \phi {x \gets \sqbr... | Let $t = \map {\operatorname{val}_\N} \tau$ be the value of $\tau$ under the natural numbers.
By definition of value of formula:
:$\N \models \map \phi {x \gets x_0}$
for every $x_0 < t$.
Therefore, {{hypothesis}}:
:$\map \phi {x \gets \sqbrk {x_0} }$
is a theorem, for every $x_0 < t$.
Additionally, by Lower Section of... | Let $\map \phi x$ be a [[Definition:WFF of Predicate Logic|WFF]] in the [[Definition:Language of Arithmetic|language of arithmetic]] with $1$ [[Definition:Free Variable (Predicate Logic)|free variable]].
Let $\tau$ be a [[Definition:Term (Predicate Logic)|term]] in the [[Definition:Language of Arithmetic|language of a... | Let $t = \map {\operatorname{val}_\N} \tau$ be the [[Definition:Value of Term under Assignment|value]] of $\tau$ under the [[Definition:Natural Number|natural numbers]].
By definition of [[Definition:Value of Formula under Assignment|value of formula]]:
:$\N \models \map \phi {x \gets x_0}$
for every $x_0 < t$.
There... | Bounded Universal Quantification of Provable Arithmetic Formula is Provable | https://proofwiki.org/wiki/Bounded_Universal_Quantification_of_Provable_Arithmetic_Formula_is_Provable | https://proofwiki.org/wiki/Bounded_Universal_Quantification_of_Provable_Arithmetic_Formula_is_Provable | [
"Natural Numbers",
"Proof Theory"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Language of Arithmetic",
"Definition:Free Variable/Predicate Logic",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Language of Arithmetic",
"Definition:Variable",
"Unary Representation of Natural Number",
... | [
"Definition:Value of Term under Assignment",
"Definition:Natural Numbers",
"Definition:Value of Formula under Assignment",
"Definition:Theorem/Formal System",
"Lower Section of Natural Number is Provable",
"Definition:Theorem/Formal System",
"False Statement implies Every Statement",
"Proof by Cases",... |
proofwiki-20761 | Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 1 | Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ have a closed locally finite refinement then:
:$T$ is paracompact. | Let every open cover of $T$ have a closed locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be a closed locally finite refinement of $\UU$, which exists by assumption.
Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$.
By definition of locally finite:
:$\forall x \... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If every [[Definition:Open Cover|open cover]] of $T$ have a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite]] [[Definition:Refinement of Cover|refinement]] then:
:$T$ is [[Definition:Paracompact Space|paracompa... | Let every [[Definition:Open Cover|open cover]] of $T$ have a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite]] [[Definition:Refinement of Cover|refinement]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $\VV$ be a [[Definition:Closed Locally Finite Set of Subsets|closed l... | Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 1 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_3_implies_Statement_1 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_3_implies_Statement_1 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Finite Set",
"Definition... |
proofwiki-20762 | Minimal Arithmetic is Sigma 1 Complete | Let $\phi$ be a $\Sigma_1$ sentence in the language of arithmetic.
Suppose $\N \models \phi$.
Then $\phi$ is a theorem in minimal arithmetic. | By definition of $\Sigma_1$, $\phi$ is logically equivalent to:
:$Q_1 Q_2 \dotsm Q_k \map \psi {x_1, x_2, \dotsc, x_k}$
where $\psi$ is a WFF with no quantifiers.
Proceed by induction on the number of quantifiers in the formula. | Let $\phi$ be a [[Definition:Arithmetical Hierarchy|$\Sigma_1$]] [[Definition:Sentence|sentence]] in the [[Definition:Language of Arithmetic|language of arithmetic]].
Suppose $\N \models \phi$.
Then $\phi$ is a [[Definition:Theorem (Formal Systems)|theorem]] in [[Definition:Minimal Arithmetic|minimal arithmetic]]. | By definition of [[Definition:Arithmetical Hierarchy|$\Sigma_1$]], $\phi$ is [[Definition:Logical Equivalence|logically equivalent]] to:
:$Q_1 Q_2 \dotsm Q_k \map \psi {x_1, x_2, \dotsc, x_k}$
where $\psi$ is a [[Definition:WFF of Predicate Logic|WFF]] with no [[Definition:Quantifier|quantifiers]].
Proceed by [[Defini... | Minimal Arithmetic is Sigma 1 Complete | https://proofwiki.org/wiki/Minimal_Arithmetic_is_Sigma_1_Complete | https://proofwiki.org/wiki/Minimal_Arithmetic_is_Sigma_1_Complete | [
"Natural Numbers",
"Proof Theory"
] | [
"Definition:Arithmetical Hierarchy",
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Arithmetic",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Definition:Arithmetical Hierarchy",
"Definition:Logical Equivalence",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Quantifier",
"Definition:Mathematical Induction",
"Definition:Quantifier",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Quantifier",
"De... |
proofwiki-20763 | Multiples of Factorial Plus One are Coprime | Let $a \in \N$ be a natural number.
Let $\sequence {x_1, x_2, \dotsc, x_n}$ be a sequence of natural numbers such that:
:$\forall i: 1 \le i \le n: x_i = 1 + i \times a!$
where $a!$ denotes the factorial of $a$.
Let $a \ge n - 1$.
Then $\set {x_i}$ are pairwise coprime. | Let $p$ be a prime number.
Suppose that $p \divides x_i$.
Then, by definition of $x_i$:
:$p \divides 1 + i \times a!$
Thus, by Consecutive Integers are Coprime:
:$p \nmid i \times a!$
Therefore:
:$p \nmid a!$
{{AimForCont}} $p \divides x_j$, where $j \ne i$.
Then $p \divides 1 + j \times a!$.
By Common Divisor Divides ... | Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
Let $\sequence {x_1, x_2, \dotsc, x_n}$ be a [[Definition:Finite Sequence|sequence]] of [[Definition:Natural Number|natural numbers]] such that:
:$\forall i: 1 \le i \le n: x_i = 1 + i \times a!$
where $a!$ denotes the [[Definition:Factorial|factorial]]... | Let $p$ be a [[Definition:Prime Number|prime number]].
Suppose that $p \divides x_i$.
Then, by definition of $x_i$:
:$p \divides 1 + i \times a!$
Thus, by [[Consecutive Integers are Coprime]]:
:$p \nmid i \times a!$
Therefore:
:$p \nmid a!$
{{AimForCont}} $p \divides x_j$, where $j \ne i$.
Then $p \divides 1 + j... | Multiples of Factorial Plus One are Coprime | https://proofwiki.org/wiki/Multiples_of_Factorial_Plus_One_are_Coprime | https://proofwiki.org/wiki/Multiples_of_Factorial_Plus_One_are_Coprime | [
"Coprime Integers",
"Factorials"
] | [
"Definition:Natural Numbers",
"Definition:Finite Sequence",
"Definition:Natural Numbers",
"Definition:Factorial",
"Definition:Pairwise Coprime/Integers"
] | [
"Definition:Prime Number",
"Consecutive Integers are Coprime",
"Common Divisor Divides Difference",
"Euclid's Lemma",
"Divisor Relation is Transitive",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Coprime Integers",
"Category:Factorials"
] |
proofwiki-20764 | Existential Quantifier Distributes over Conjunction | Let $\map \phi x, \map \psi x$ be WFFs of the free variable $x$.
Then:
:$\exists x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\exists x: \map \phi x} \land \paren {\exists x: \map \psi x}$ | {{BeginTableau|\exists x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\exists x: \map \phi x} \land \paren {\exists x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \exists x: \paren {\map \phi x \land \map \psi x}
|rlnk = Rule of Assumption
|rtxt = Pre... | Let $\map \phi x, \map \psi x$ be [[Definition:WFF of Predicate Logic|WFFs]] of the [[Definition:Free Variable|free variable]] $x$.
Then:
:$\exists x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\exists x: \map \phi x} \land \paren {\exists x: \map \psi x}$ | {{BeginTableau|\exists x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\exists x: \map \phi x} \land \paren {\exists x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \exists x: \paren {\map \phi x \land \map \psi x}
|rlnk = Rule of Assumption
|rtxt = Pre... | Existential Quantifier Distributes over Conjunction | https://proofwiki.org/wiki/Existential_Quantifier_Distributes_over_Conjunction | https://proofwiki.org/wiki/Existential_Quantifier_Distributes_over_Conjunction | [
"Existential Quantifier"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable"
] | [
"Category:Existential Quantifier"
] |
proofwiki-20765 | Universal Quantifier Distributes over Conjunction | Let $\map \phi x, \map \psi x$ be WFFs of the free variable $x$.
Then:
:$\forall x: \paren {\map \phi x \land \map \psi x} \dashv \vdash \paren {\forall x: \map \phi x} \land \paren {\forall x: \map \psi x}$ | === Forward Implication ===
{{BeginTableau|\forall x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\forall x: \map \phi x} \land \paren {\forall x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \forall x: \paren {\map \phi x \land \map \psi x}
|rlnk = Rule of Assumpt... | Let $\map \phi x, \map \psi x$ be [[Definition:WFF of Predicate Logic|WFFs]] of the [[Definition:Free Variable|free variable]] $x$.
Then:
:$\forall x: \paren {\map \phi x \land \map \psi x} \dashv \vdash \paren {\forall x: \map \phi x} \land \paren {\forall x: \map \psi x}$ | === Forward Implication ===
{{BeginTableau|\forall x: \paren {\map \phi x \land \map \psi x} \vdash \paren {\forall x: \map \phi x} \land \paren {\forall x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \forall x: \paren {\map \phi x \land \map \psi x}
|rlnk = Rule of Assump... | Universal Quantifier Distributes over Conjunction | https://proofwiki.org/wiki/Universal_Quantifier_Distributes_over_Conjunction | https://proofwiki.org/wiki/Universal_Quantifier_Distributes_over_Conjunction | [
"Universal Quantifier"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable"
] | [] |
proofwiki-20766 | Existential Quantifier Distributes over Disjunction | Let $\map \phi x, \map \psi x$ be WFFs of the free variable $x$.
Then:
:$\exists x: \paren {\map \phi x \lor \map \psi x} \dashv \vdash \paren {\exists x: \map \phi x} \lor \paren {\exists x: \map \psi x}$ | === Forward Implication ===
{{BeginTableau|\exists x: \paren {\map \phi x \lor \map \psi x} \vdash \paren {\exists x: \map \phi x} \lor \paren {\exists x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \exists x: \paren {\map \phi x \lor \map \psi x}
|rlnk = Rule of Assumption... | Let $\map \phi x, \map \psi x$ be [[Definition:WFF of Predicate Logic|WFFs]] of the [[Definition:Free Variable|free variable]] $x$.
Then:
:$\exists x: \paren {\map \phi x \lor \map \psi x} \dashv \vdash \paren {\exists x: \map \phi x} \lor \paren {\exists x: \map \psi x}$ | === Forward Implication ===
{{BeginTableau|\exists x: \paren {\map \phi x \lor \map \psi x} \vdash \paren {\exists x: \map \phi x} \lor \paren {\exists x: \map \psi x} }}
{{TableauLine|n = 1
|pool = 1
|f = \exists x: \paren {\map \phi x \lor \map \psi x}
|rlnk = Rule of Assumptio... | Existential Quantifier Distributes over Disjunction | https://proofwiki.org/wiki/Existential_Quantifier_Distributes_over_Disjunction | https://proofwiki.org/wiki/Existential_Quantifier_Distributes_over_Disjunction | [
"Existential Quantifier"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable"
] | [] |
proofwiki-20767 | Universal Quantifier Distributes over Disjunction | Let $\map \phi x, \map \psi x$ be WFFs of the free variable $x$.
Then:
:$\paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x} \vdash \forall x: \paren {\map \phi x \lor \map \psi x}$ | {{BeginTableau | \paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x} \vdash \forall x: \paren {\map \phi x \lor \map \psi x}
}}
{{TableauLine | n = 1
| pool = 1
| f = \paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x}
| rlnk = Rule of Assumpti... | Let $\map \phi x, \map \psi x$ be [[Definition:WFF of Predicate Logic|WFFs]] of the [[Definition:Free Variable|free variable]] $x$.
Then:
:$\paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x} \vdash \forall x: \paren {\map \phi x \lor \map \psi x}$ | {{BeginTableau | \paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x} \vdash \forall x: \paren {\map \phi x \lor \map \psi x}
}}
{{TableauLine | n = 1
| pool = 1
| f = \paren {\forall x: \map \phi x} \lor \paren {\forall x: \map \psi x}
| rlnk = Rule of Assumpti... | Universal Quantifier Distributes over Disjunction | https://proofwiki.org/wiki/Universal_Quantifier_Distributes_over_Disjunction | https://proofwiki.org/wiki/Universal_Quantifier_Distributes_over_Disjunction | [
"Universal Quantifier"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable"
] | [
"Category:Universal Quantifier"
] |
proofwiki-20768 | Characterization of Paracompactness in T3 Space/Lemma 5 | :$\forall V \in \VV: V \subseteq V^*$ | Let $V \in \VV$.
Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.
From Subset of Set Difference iff Disjoint Set:
:$\forall A \in \AA_V : V \subseteq X \setminus A$
We have:
{{begin-eqn}}
{{eqn | l = V
| o = \subseteq
| r = \bigcap \set{X \setminus A : A \in \AA_V}
| c = Set is Subset of Intersection of... | :$\forall V \in \VV: V \subseteq V^*$ | Let $V \in \VV$.
Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.
From [[Subset of Set Difference iff Disjoint Set]]:
:$\forall A \in \AA_V : V \subseteq X \setminus A$
We have:
{{begin-eqn}}
{{eqn | l = V
| o = \subseteq
| r = \bigcap \set{X \setminus A : A \in \AA_V}
| c = [[Set is Subset of Int... | Characterization of Paracompactness in T3 Space/Lemma 5 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_5 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_5 | [
"Characterization of Paracompactness in T3 Space"
] | [] | [
"Subset of Set Difference iff Disjoint Set",
"Set is Subset of Intersection of Supersets",
"De Morgan's Laws (Set Theory)/Set Difference",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20769 | Basic Primitive Recursive Functions are Arithmetically Definable | Let $f: \N^k \to \N$ be a basic primitive recursive function.
Then there is a WFF $\map \phi {y, x_1, \dotsc, x_k}$ of $k + 1$ free variables and no quantifiers such that:
:$y = \map f {x_1, \dotsc, x_k}$
{{iff}}
:$\N \models \map \phi {y \gets \sqbrk y, x_1 \gets \sqbrk {x_1}, \dotsc, x_k \gets \sqbrk {x_k} }$
where $... | === Zero Function ===
Suppose $\map f x = 0$.
Then:
:$\map \phi {y, x} := y = 0$
Correctness is apparent.
{{qed|lemma}} | Let $f: \N^k \to \N$ be a [[Definition:Basic Primitive Recursive Function|basic primitive recursive function]].
Then there is a [[Definition:WFF of Predicate Logic|WFF]] $\map \phi {y, x_1, \dotsc, x_k}$ of $k + 1$ [[Definition:Free Variable|free variables]] and no [[Definition:Quantifier|quantifiers]] such that:
:$y ... | === Zero Function ===
Suppose $\map f x = 0$.
Then:
:$\map \phi {y, x} := y = 0$
Correctness is apparent.
{{qed|lemma}} | Basic Primitive Recursive Functions are Arithmetically Definable | https://proofwiki.org/wiki/Basic_Primitive_Recursive_Functions_are_Arithmetically_Definable | https://proofwiki.org/wiki/Basic_Primitive_Recursive_Functions_are_Arithmetically_Definable | [
"Primitive Recursive Functions"
] | [
"Definition:Basic Primitive Recursive Function",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Definition:Quantifier",
"Unary Representation of Natural Number"
] | [] |
proofwiki-20770 | Substitution of Arithmetically Definable Functions is Arithmetically Definable | Let $f: \N^t \to \N$ and $g_1, g_2, \dotsc, g_t: \N^k \to \N$ be partial functions.
Let $h: \N^k \to \N$ be defined by substitution as:
:$\map h {x_1, \dotsc, x_k} \approx \map f {\map {g_1} {x_1, \dotsc, x_k}, \dotsc, \map {g_t} {x_1, \dotsc, x_k} }$
Suppose that there exists a $\Sigma_1$ WFF of $t + 1$ free variable... | Define:
:$\map {\phi_h} {y, x_1, \dotsc, x_k} := \exists s_1: \dotsm \exists s_t: \map {\phi_f} {y, s_1, \dotsc, s_t} \land \map {\phi_{g_1} } {s_1, x_1, \dotsc, x_k} \land \dotso \land \map {\phi_{g_t} } {s_t, x_1, \dotsc, x_k}$
Correctness is apparent.
{{TheoremWanted|$\paren {\exists x: \map \phi x} \land \psi \vdas... | Let $f: \N^t \to \N$ and $g_1, g_2, \dotsc, g_t: \N^k \to \N$ be [[Definition:Partial Function|partial functions]].
Let $h: \N^k \to \N$ be defined by [[Definition:Substitution (Mathematical Logic)|substitution]] as:
:$\map h {x_1, \dotsc, x_k} \approx \map f {\map {g_1} {x_1, \dotsc, x_k}, \dotsc, \map {g_t} {x_1, \d... | Define:
:$\map {\phi_h} {y, x_1, \dotsc, x_k} := \exists s_1: \dotsm \exists s_t: \map {\phi_f} {y, s_1, \dotsc, s_t} \land \map {\phi_{g_1} } {s_1, x_1, \dotsc, x_k} \land \dotso \land \map {\phi_{g_t} } {s_t, x_1, \dotsc, x_k}$
Correctness is apparent.
{{TheoremWanted|$\paren {\exists x: \map \phi x} \land \psi \vd... | Substitution of Arithmetically Definable Functions is Arithmetically Definable | https://proofwiki.org/wiki/Substitution_of_Arithmetically_Definable_Functions_is_Arithmetically_Definable | https://proofwiki.org/wiki/Substitution_of_Arithmetically_Definable_Functions_is_Arithmetically_Definable | [
"Primitive Recursive Functions"
] | [
"Definition:Partial Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number",
"Definition:Arithmetical Hierarchy",
"Definition:Language o... | [
"Definition:Arithmetical Hierarchy",
"Conjunction of Existential Quantifier",
"Category:Primitive Recursive Functions"
] |
proofwiki-20771 | Minimization of Arithmetically Definable Function is Arithmetically Definable | Let $f: \N^{k + 1} \to \N$ be a partial function.
Let $g: \N^k \to \N$ be the partial function defined as:
:$\map g {x_1, \dotsc, x_k} \approx \map {\mu z} {\map f {x_1, \dotsc, x_k, z} }$
where $\mu$ is minimization.
Suppose that there exists a $\Sigma_1$ WFF of $k + 2$ free variables:
:$\map {\phi_f} {y, x_1, \dotsc,... | Define:
:$\map {\phi_g} {y, x_1, \dotsc, x_k} := \map {\phi_f} {0, x_1, \dotsc, x_k, y} \land \forall p < y: \exists q: \paren {q \ne 0 \land \map {\phi_f} {q, x_1, \dotsc, x_k, y} }$
For, by definition of minimization:
:$y = \map g {x_1, \dotsc, x_k}$
{{iff}}
:$\map f {x_1, \dotsc, x_k, y} = 0$
and
:$\map f {x_1, \dot... | Let $f: \N^{k + 1} \to \N$ be a [[Definition:Total Function|partial function]].
Let $g: \N^k \to \N$ be the [[Definition:Partial Function|partial function]] defined as:
:$\map g {x_1, \dotsc, x_k} \approx \map {\mu z} {\map f {x_1, \dotsc, x_k, z} }$
where $\mu$ is [[Definition:Minimization|minimization]].
Suppose th... | Define:
:$\map {\phi_g} {y, x_1, \dotsc, x_k} := \map {\phi_f} {0, x_1, \dotsc, x_k, y} \land \forall p < y: \exists q: \paren {q \ne 0 \land \map {\phi_f} {q, x_1, \dotsc, x_k, y} }$
For, by definition of [[Definition:Minimization|minimization]]:
:$y = \map g {x_1, \dotsc, x_k}$
{{iff}}
:$\map f {x_1, \dotsc, x_k, y... | Minimization of Arithmetically Definable Function is Arithmetically Definable | https://proofwiki.org/wiki/Minimization_of_Arithmetically_Definable_Function_is_Arithmetically_Definable | https://proofwiki.org/wiki/Minimization_of_Arithmetically_Definable_Function_is_Arithmetically_Definable | [
"Recursive Functions"
] | [
"Definition:Total Function",
"Definition:Partial Function",
"Definition:Minimization",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number",
"Definition:Arithmetical Hierarchy",
"Definition:... | [
"Definition:Minimization",
"Definition:Arithmetical Hierarchy",
"Conjunction of Existential Quantifier",
"Bounded Universal Quantifier Distributes over Conjunction",
"Category:Recursive Functions"
] |
proofwiki-20772 | Addition is Arithmetically Definable | Let $f: \N^2 \to \N$ be defined as:
:$\map f {x, y} = x + y$
Then there exists a WFF $\map \phi {z, x, y}$ of $3$ free variables and containing no quantifiers such that:
:$z = \map f {x, y} \iff \N \models \map \phi {\sqbrk z, \sqbrk x, \sqbrk y}$
where $\sqbrk a$ is the unary representation of $a \in \N$. | Define:
:$\map \phi {z, x, y} := z = x + y$
Correctness is trivial.
{{qed}}
Category:Natural Number Addition
lamehfpue8qwsjw4b4v78lr8rsi7cyl | Let $f: \N^2 \to \N$ be defined as:
:$\map f {x, y} = x + y$
Then there exists a [[Definition:WFF of Predicate Logic|WFF]] $\map \phi {z, x, y}$ of $3$ [[Definition:Free Variable|free variables]] and containing no [[Definition:Quantifier|quantifiers]] such that:
:$z = \map f {x, y} \iff \N \models \map \phi {\sqbrk z,... | Define:
:$\map \phi {z, x, y} := z = x + y$
Correctness is trivial.
{{qed}}
[[Category:Natural Number Addition]]
lamehfpue8qwsjw4b4v78lr8rsi7cyl | Addition is Arithmetically Definable | https://proofwiki.org/wiki/Addition_is_Arithmetically_Definable | https://proofwiki.org/wiki/Addition_is_Arithmetically_Definable | [
"Natural Number Addition"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Definition:Quantifier",
"Unary Representation of Natural Number"
] | [
"Category:Natural Number Addition"
] |
proofwiki-20773 | Multiplication is Arithmetically Definable | Let $f: \N^2 \to \N$ be defined as:
:$\map f {x, y} = x \times y$
Then there exists a WFF $\map \phi {z, x, y}$ of $3$ free variables and containing no quantifiers such that:
:$z = \map f {x, y} \iff \N \models \map \phi {\sqbrk z, \sqbrk x, \sqbrk y}$
where $\sqbrk a$ is the unary representation of $a \in \N$. | Define:
:$\map \phi {z, x, y} := z = x \times y$
Correctness is trivial.
{{qed}}
Category:Natural Number Multiplication
fvi8hhh30ljc9u6hn7oi243p1kjf531 | Let $f: \N^2 \to \N$ be defined as:
:$\map f {x, y} = x \times y$
Then there exists a [[Definition:WFF of Predicate Logic|WFF]] $\map \phi {z, x, y}$ of $3$ [[Definition:Free Variable|free variables]] and containing no [[Definition:Quantifier|quantifiers]] such that:
:$z = \map f {x, y} \iff \N \models \map \phi {\sqb... | Define:
:$\map \phi {z, x, y} := z = x \times y$
Correctness is trivial.
{{qed}}
[[Category:Natural Number Multiplication]]
fvi8hhh30ljc9u6hn7oi243p1kjf531 | Multiplication is Arithmetically Definable | https://proofwiki.org/wiki/Multiplication_is_Arithmetically_Definable | https://proofwiki.org/wiki/Multiplication_is_Arithmetically_Definable | [
"Natural Number Multiplication"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Definition:Quantifier",
"Unary Representation of Natural Number"
] | [
"Category:Natural Number Multiplication"
] |
proofwiki-20774 | Remainder is Arithmetically Definable | Let $\operatorname{rem}: \N^2 \to \N$ be defined as:
:$\map \rem {n, m} = \begin{cases}
\text{the remainder when } n \text{ is divided by } m & : m \ne 0 \\
0 & : m = 0
\end{cases}$
where the $\text{remainder}$ is as defined in the Division Theorem:
:If $n = m q + r$, where $0 \le r < m$, then $r$ is the remainder.
The... | Define:
:$\map \phi {r, n, m} := \paren {m = 0 \land r = 0} \lor \paren {m \ne 0 \land r < m \land \exists q: n = \paren {m \times q} + r}$
Suppose $m = 0$.
Then, only the first case:
:$m = 0 \land r = 0$
can possibly hold.
Therefore, the formula holds {{iff}} $r = 0$, which matches the definition.
Suppose $m \ne 0$.
T... | Let $\operatorname{rem}: \N^2 \to \N$ be defined as:
:$\map \rem {n, m} = \begin{cases}
\text{the remainder when } n \text{ is divided by } m & : m \ne 0 \\
0 & : m = 0
\end{cases}$
where the $\text{remainder}$ is as defined in the [[Division Theorem]]:
:If $n = m q + r$, where $0 \le r < m$, then $r$ is the [[Definiti... | Define:
:$\map \phi {r, n, m} := \paren {m = 0 \land r = 0} \lor \paren {m \ne 0 \land r < m \land \exists q: n = \paren {m \times q} + r}$
Suppose $m = 0$.
Then, only the first case:
:$m = 0 \land r = 0$
can possibly hold.
Therefore, the formula holds {{iff}} $r = 0$, which matches the definition.
Suppose $m \ne... | Remainder is Arithmetically Definable | https://proofwiki.org/wiki/Remainder_is_Arithmetically_Definable | https://proofwiki.org/wiki/Remainder_is_Arithmetically_Definable | [] | [
"Division Theorem",
"Definition:Remainder",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number"
] | [
"Conjunction of Existential Quantifier",
"Existential Quantifier Distributes over Disjunction"
] |
proofwiki-20775 | Gödel's Beta Function is Arithmetically Definable | Let Gödel's $\beta$ function $\beta: \N^3 \to \N$ be defined as:
:$\map \beta {x, y, z} = \map \rem {x, 1 + \paren {z + 1} \times y}$
Then there exists a $\Sigma_1$ WFF of $4$ free variables:
:$\map \phi {r, x, y, z}$
such that:
:$r = \map \beta {x, y, z} \iff \N \models \map \phi {\sqbrk r, \sqbrk x, \sqbrk y, \sqbrk ... | Follows from:
:Basic Primitive Recursive Functions are Arithmetically Definable
:Addition is Arithmetically Definable
:Multiplication is Arithmetically Definable
:Remainder is Arithmetically Definable
:Substitution of Arithmetically Definable Functions is Arithmetically Definable
{{qed}}
3t1yaqraz24jma4sowyqrblnpw5u1pg | Let [[Definition:Gödel's Beta Function|Gödel's $\beta$ function]] $\beta: \N^3 \to \N$ be defined as:
:$\map \beta {x, y, z} = \map \rem {x, 1 + \paren {z + 1} \times y}$
Then there exists a [[Definition:Arithmetical Hierarchy|$\Sigma_1$]] [[Definition:WFF of Predicate Logic|WFF]] of $4$ [[Definition:Free Variable|fre... | Follows from:
:[[Basic Primitive Recursive Functions are Arithmetically Definable]]
:[[Addition is Arithmetically Definable]]
:[[Multiplication is Arithmetically Definable]]
:[[Remainder is Arithmetically Definable]]
:[[Substitution of Arithmetically Definable Functions is Arithmetically Definable]]
{{qed}}
3t1yaqraz24... | Gödel's Beta Function is Arithmetically Definable | https://proofwiki.org/wiki/Gödel's_Beta_Function_is_Arithmetically_Definable | https://proofwiki.org/wiki/Gödel's_Beta_Function_is_Arithmetically_Definable | [] | [
"Definition:Gödel's Beta Function",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number"
] | [
"Basic Primitive Recursive Functions are Arithmetically Definable",
"Addition is Arithmetically Definable",
"Multiplication is Arithmetically Definable",
"Remainder is Arithmetically Definable",
"Substitution of Arithmetically Definable Functions is Arithmetically Definable"
] |
proofwiki-20776 | Reduction Formula for Primitive of Product of Power with Exponential | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let:
:$I_n := \ds \int x^n e^x \rd x$
Then:
:$I_n = x^n e^x - n I_{n - 1}$
is a reduction formula for $\ds \int x^n e^x \rd x$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = Power Rule for Derivatives
}}
{{... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let:
:$I_n := \ds \int x^n e^x \rd x$
Then:
:$I_n = x^n e^x - n I_{n - 1}$
is a [[Definition:Reduction Formula (Calculus)|reduction formula]] for $\ds \int x^n e^x \rd x$. | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
|... | Reduction Formula for Primitive of Product of Power with Exponential | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_Product_of_Power_with_Exponential | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_Product_of_Power_with_Exponential | [
"Reduction Formulae (Calculus)",
"Primitives involving Exponential Function"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Reduction Formula (Calculus)"
] | [
"Definition:Primitive (Calculus)",
"Power Rule for Derivatives",
"Derivative of Exponential Function",
"Integration by Parts"
] |
proofwiki-20777 | Subset of Cover is Cover of Subset | Let $S$ be a set.
Let $\CC$ be a cover of $S$.
Let $T \subseteq S$.
Let:
:$\CC_T = \set {C \in \CC : C \cap T \ne \O}$
Then $\CC_T$ is a cover of $T$:
:$T \subseteq \bigcup \CC_T$ | Let $x \in T$.
By definition of a cover:
:$\exists C \in \CC : x \in C$
By definition of set intersection:
:$x \in C \cap T$
Hence:
:$C \in \CC_T$
Since $x$ was arbitrary, it follows that $\CC_T$ is a cover of $T$ by definition and:
:$T \subseteq \bigcup \CC_T$
{{qed}}
Category:Covers
Category:Subsets
o4iohi477mtx4na6q... | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] of $S$.
Let $T \subseteq S$.
Let:
:$\CC_T = \set {C \in \CC : C \cap T \ne \O}$
Then $\CC_T$ is a [[Definition:Cover of Set|cover]] of $T$:
:$T \subseteq \bigcup \CC_T$ | Let $x \in T$.
By definition of a [[Definition:Cover of Set|cover]]:
:$\exists C \in \CC : x \in C$
By definition of [[Definition:Set Intersection|set intersection]]:
:$x \in C \cap T$
Hence:
:$C \in \CC_T$
Since $x$ was arbitrary, it follows that $\CC_T$ is a [[Definition:Cover of Set|cover]] of $T$ by definition... | Subset of Cover is Cover of Subset | https://proofwiki.org/wiki/Subset_of_Cover_is_Cover_of_Subset | https://proofwiki.org/wiki/Subset_of_Cover_is_Cover_of_Subset | [
"Covers",
"Subsets"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Cover of Set"
] | [
"Definition:Cover of Set",
"Definition:Set Intersection",
"Definition:Cover of Set",
"Category:Covers",
"Category:Subsets"
] |
proofwiki-20778 | Gödel's Beta Function Lemma | Let $\beta: \N^3 \to \N$ be Gödel's $\beta$ function.
Let $\sequence {x_0, x_1, \dotsc, x_n }$ be a finite sequence of natural numbers.
Then there exist some $a, b \in \N$ such that, for every $i \le n$:
:$\map \beta {a, b, i} = x_i$ | Let $M = \map \max {n, x_0, x_1, \dotsc, x_n}$.
Define $b = M!$, the factorial of $M$.
Let $\sequence {y_1, y_2, \dotsc, y_{n + 1} }$ be the finite sequence defined by:
:$y_i = 1 + \paren {i \times b}$
As $M \ge n$, by definition:
:$M \ge \paren {n + 1} - 1$
Therefore, by Multiples of Factorial Plus One are Coprime, th... | Let $\beta: \N^3 \to \N$ be [[Definition:Gödel's Beta Function|Gödel's $\beta$ function]].
Let $\sequence {x_0, x_1, \dotsc, x_n }$ be a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Natural Number|natural numbers]].
Then there exist some $a, b \in \N$ such that, for every $i \le n$:
:$\map \beta {a,... | Let $M = \map \max {n, x_0, x_1, \dotsc, x_n}$.
Define $b = M!$, the [[Definition:Factorial|factorial]] of $M$.
Let $\sequence {y_1, y_2, \dotsc, y_{n + 1} }$ be the [[Definition:Finite Sequence|finite sequence]] defined by:
:$y_i = 1 + \paren {i \times b}$
As $M \ge n$, by definition:
:$M \ge \paren {n + 1} - 1$
T... | Gödel's Beta Function Lemma | https://proofwiki.org/wiki/Gödel's_Beta_Function_Lemma | https://proofwiki.org/wiki/Gödel's_Beta_Function_Lemma | [
"Gödel's Beta Function"
] | [
"Definition:Gödel's Beta Function",
"Definition:Finite Sequence",
"Definition:Natural Numbers"
] | [
"Definition:Factorial",
"Definition:Finite Sequence",
"Multiples of Factorial Plus One are Coprime",
"Definition:Pairwise Coprime/Integers",
"Chinese Remainder Theorem",
"Definition:Gödel's Beta Function",
"Category:Gödel's Beta Function"
] |
proofwiki-20779 | Primitive Recursion on Arithmetically Definable Function is Arithmetically Definable | Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be partial functions.
Let $h: \N^{k + 1} \to \N$ be obtained from $f$ and $g$ by primitive recursion.
Let there exist $\Sigma_1$ WFFs:
:$\map {\phi_f} {y, x_1, \dotsc, x_k}$
:$\map {\phi_g} {y, x_1, \dotsc, x_k, n, z}$
of $k + 1$ and $k + 3$ free variables, respectively, su... | By Gödel's Beta Function is Arithmetically Definable, let:
:$\map {\phi_\beta} {y, a, b, i}$
be a $\Sigma_1$ WFF of $4$ free variables such that:
$y = \map \beta {a, b, i} \iff \N \models \map {\phi_\beta} {\sqbrk y, \sqbrk a, \sqbrk b, \sqbrk i}$
Define $\map {\phi_h} {y, x_1, \dotsc, x_k, n}$ as:
:$\exists a: \exists... | Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be [[Definition:Partial Function|partial functions]].
Let $h: \N^{k + 1} \to \N$ be obtained from $f$ and $g$ by [[Definition:Primitive Recursion|primitive recursion]].
Let there exist [[Definition:Arithmetical Hierarchy|$\Sigma_1$]] [[Definition:WFF of Predicate Logic|W... | By [[Gödel's Beta Function is Arithmetically Definable]], let:
:$\map {\phi_\beta} {y, a, b, i}$
be a [[Definition:Arithmetical Hierarchy|$\Sigma_1$]] [[Definition:WFF of Predicate Logic|WFF]] of $4$ [[Definition:Free Variable|free variables]] such that:
$y = \map \beta {a, b, i} \iff \N \models \map {\phi_\beta} {\sqb... | Primitive Recursion on Arithmetically Definable Function is Arithmetically Definable | https://proofwiki.org/wiki/Primitive_Recursion_on_Arithmetically_Definable_Function_is_Arithmetically_Definable | https://proofwiki.org/wiki/Primitive_Recursion_on_Arithmetically_Definable_Function_is_Arithmetically_Definable | [
"Primitive Recursive Functions"
] | [
"Definition:Partial Function",
"Definition:Primitive Recursion",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Lo... | [
"Gödel's Beta Function is Arithmetically Definable",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Definition:Finite Sequence",
"Definition:Function",
"Definition:Free Variable",
"Definition:Free Variable",
"Gödel's Beta Fu... |
proofwiki-20780 | Reduction Formula for Primitive of Product of Power with Power of Quadratic | Let $n \in \Z_{\ge 0}$ and $k \in \Z_{\ge 2}$.
Let:
:$I_{n, k} := \ds \int x^k \paren {x^2 + A x + B}^n \rd x$
Then:
:$I_{n, k} = \dfrac {x^{k - 1} \paren {x^2 + A x + B}^{n + 1} } {k + 2 n + 1} - \dfrac {B \paren {k - 1} } {k + 2 n + 1} I_{n, k - 2} - \dfrac {A \paren {k + n} } {k + 2 n + 1} I_{n, k - 1}$
is a reduct... | Let $h$ be the real function defined as:
:$\forall x \in \R: \map h x = x^2 + A x + B$
Thus we have:
:$I_{n, k} := \ds \int x^k \paren {\map h x}^n \rd x$
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {x^{k - 1} \paren {\map h x}^{n + 1} }
| r = x^{k - 1} \map {\dfrac \d {\d x} } {\paren {\map ... | Let $n \in \Z_{\ge 0}$ and $k \in \Z_{\ge 2}$.
Let:
:$I_{n, k} := \ds \int x^k \paren {x^2 + A x + B}^n \rd x$
Then:
:$I_{n, k} = \dfrac {x^{k - 1} \paren {x^2 + A x + B}^{n + 1} } {k + 2 n + 1} - \dfrac {B \paren {k - 1} } {k + 2 n + 1} I_{n, k - 2} - \dfrac {A \paren {k + n} } {k + 2 n + 1} I_{n, k - 1}$
is a [[De... | Let $h$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map h x = x^2 + A x + B$
Thus we have:
:$I_{n, k} := \ds \int x^k \paren {\map h x}^n \rd x$
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {x^{k - 1} \paren {\map h x}^{n + 1} }
| r = x^{k - 1} \map ... | Reduction Formula for Primitive of Product of Power with Power of Quadratic | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_Product_of_Power_with_Power_of_Quadratic | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_Product_of_Power_with_Power_of_Quadratic | [
"Primitives involving a x squared plus b x plus c",
"Reduction Formulae (Calculus)"
] | [
"Definition:Reduction Formula (Calculus)"
] | [
"Definition:Real Function",
"Product Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-20781 | Reduction Formula for Primitive of a x + b over Power of Quadratic | Let $n \in \Z_{\ge 2}$.
Let:
:$\map {I_n} {a, b} := \ds \int \dfrac {a x + b} {\paren {x^2 + A x + B}^n} \rd x$
Then:
:$\map {I_n} {a, b} = \dfrac {b A - 2 a B + \paren {2 b - a A} x} {\paren {n - 1} \paren {4 B - A^2} \paren {x^2 + A x + B}^n} + \dfrac {\paren {2 n - 3} \paren {2 b - a A} } {\paren {n - 1} \paren {4 B... | We observe that:
:$(1): \quad \map {\dfrac \d {\d x} } {x^2 + A x + B} = 2 x + A$
Hence we obtain:
{{begin-eqn}}
{{eqn | l = a x + b
| r = \dfrac a 2 \paren {2 x + A - A} + b
| c =
}}
{{eqn | r = \dfrac a 2 \paren {2 x + A} + \paren {b - \frac {a A} 2}
| c =
}}
{{end-eqn}}
and so express:
{{begin-eq... | Let $n \in \Z_{\ge 2}$.
Let:
:$\map {I_n} {a, b} := \ds \int \dfrac {a x + b} {\paren {x^2 + A x + B}^n} \rd x$
Then:
:$\map {I_n} {a, b} = \dfrac {b A - 2 a B + \paren {2 b - a A} x} {\paren {n - 1} \paren {4 B - A^2} \paren {x^2 + A x + B}^n} + \dfrac {\paren {2 n - 3} \paren {2 b - a A} } {\paren {n - 1} \paren {4... | We observe that:
:$(1): \quad \map {\dfrac \d {\d x} } {x^2 + A x + B} = 2 x + A$
Hence we obtain:
{{begin-eqn}}
{{eqn | l = a x + b
| r = \dfrac a 2 \paren {2 x + A - A} + b
| c =
}}
{{eqn | r = \dfrac a 2 \paren {2 x + A} + \paren {b - \frac {a A} 2}
| c =
}}
{{end-eqn}}
and so express:
{{be... | Reduction Formula for Primitive of a x + b over Power of Quadratic | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_a_x_+_b_over_Power_of_Quadratic | https://proofwiki.org/wiki/Reduction_Formula_for_Primitive_of_a_x_+_b_over_Power_of_Quadratic | [
"Primitives involving a x squared plus b x plus c",
"Reduction Formulae (Calculus)"
] | [
"Definition:Reduction Formula (Calculus)"
] | [
"Integration by Substitution",
"Primitive of Power",
"Definition:Real Function",
"Power Rule for Derivatives",
"Product Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Definition:Fraction/Numerator",
"Definition:Integration/In... |
proofwiki-20782 | Characterization of Paracompactness in T3 Space/Lemma 11 | :$\forall A \in \AA, V^* \in \VV^* : A \cap V^* \ne \O \implies A \cap V \ne \O$ | We prove the contrapositive statement:
:$\forall A \in \AA, V \in \VV^* : A \cap V = \O \implies A \cap V^* = \O$
Let $B \in \AA, V^* \in \VV^* : B \cap V = \O$.
Hence:
:$B \in \set{A \in \AA : A \cap V = \O }$
From Set is Subset of Union:
:$B \subseteq \ds \bigcup \set{A \in \AA : A \cap V = \O }$
We have:
{{begin-eqn... | :$\forall A \in \AA, V^* \in \VV^* : A \cap V^* \ne \O \implies A \cap V \ne \O$ | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$\forall A \in \AA, V \in \VV^* : A \cap V = \O \implies A \cap V^* = \O$
Let $B \in \AA, V^* \in \VV^* : B \cap V = \O$.
Hence:
:$B \in \set{A \in \AA : A \cap V = \O }$
From [[Set is Subset of Union]]:
:$B \subseteq \ds \bigcup \set... | Characterization of Paracompactness in T3 Space/Lemma 11 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_11 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_11 | [
"Characterization of Paracompactness in T3 Space"
] | [] | [
"Definition:Contrapositive Statement",
"Set is Subset of Union",
"Set Difference with Subset is Superset of Set Difference",
"Subset of Set Difference iff Disjoint Set",
"Rule of Transposition",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20783 | Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 6 | Let $T = \struct {X, \tau}$ be a topological space.
If $T$ is paracompact then:
:every open cover of $T$ has an open $\sigma$-locally finite refinement | {{Recall|Paracompact Space}}
{{:Definition:Paracompact Space}}
From Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets:
:every open cover of $T$ has an open $\sigma$-locally finite refinement. | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
If $T$ is [[Definition:Paracompact Space|paracompact]] then:
:every [[Definition:Open Cover|open cover]] of $T$ has an [[Definition:Open Sigma-Locally Finite Set of Subsets|open $\sigma$-locally finite]] [[Definition:Refinement of Co... | {{Recall|Paracompact Space}}
{{:Definition:Paracompact Space}}
From [[Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets]]:
:every [[Definition:Open Cover|open cover]] of $T$ has an [[Definition:Open Sigma-Locally Finite Set of Subsets|open $\sigma$-locally finite]] [[Definition:Refinement of Cover|r... | Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_1_implies_Statement_6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_1_implies_Statement_6 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Paracompact Space",
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets",
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] |
proofwiki-20784 | Discrete Set of Subsets is Locally Finite | Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF$ be a discrete set of subsets of $S$.
Then $\FF$ is a locally finite set of subsets of $S$. | {{Recall|Discrete Set of Subsets|discrete set of subsets}}
{{:Definition:Discrete Set of Subsets}}
{{Recall|Locally Finite Set of Subsets|locally finite set of subsets}}
{{:Definition:Locally Finite Set of Subsets}}
The result follows immediately.
{{qed}}
Category:Discrete Sets of Subsets
Category:Locally Finite Sets o... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Discrete Set of Subsets|discrete set of subsets]] of $S$.
Then $\FF$ is a [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]] of $S$. | {{Recall|Discrete Set of Subsets|discrete set of subsets}}
{{:Definition:Discrete Set of Subsets}}
{{Recall|Locally Finite Set of Subsets|locally finite set of subsets}}
{{:Definition:Locally Finite Set of Subsets}}
The result follows immediately.
{{qed}}
[[Category:Discrete Sets of Subsets]]
[[Category:Locally Fini... | Discrete Set of Subsets is Locally Finite | https://proofwiki.org/wiki/Discrete_Set_of_Subsets_is_Locally_Finite | https://proofwiki.org/wiki/Discrete_Set_of_Subsets_is_Locally_Finite | [
"Discrete Sets of Subsets",
"Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Discrete Set of Subsets",
"Definition:Locally Finite Set of Subsets"
] | [
"Category:Discrete Sets of Subsets",
"Category:Locally Finite Sets of Subsets"
] |
proofwiki-20785 | Sigma-Discrete Set of Subsets is Sigma-Locally Finite | Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF$ be a $\sigma$-discrete set of subsets of $S$.
Then $\FF$ is a $\sigma$-locally finite set of subsets of $S$. | {{Recall|Sigma-Discrete Set of Subsets|$\sigma$-discrete set of subsets}}
{{:Definition:Sigma-Discrete Set of Subsets}}
{{Recall|Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite set of subsets}}
{{:Definition:Sigma-Locally Finite Set of Subsets}}
The result follows immediately from Discrete Set of Subsets is... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Sigma-Discrete Set of Subsets|$\sigma$-discrete set of subsets]] of $S$.
Then $\FF$ is a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite set of subsets]] of $S$. | {{Recall|Sigma-Discrete Set of Subsets|$\sigma$-discrete set of subsets}}
{{:Definition:Sigma-Discrete Set of Subsets}}
{{Recall|Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite set of subsets}}
{{:Definition:Sigma-Locally Finite Set of Subsets}}
The result follows immediately from [[Discrete Set of Subset... | Sigma-Discrete Set of Subsets is Sigma-Locally Finite | https://proofwiki.org/wiki/Sigma-Discrete_Set_of_Subsets_is_Sigma-Locally_Finite | https://proofwiki.org/wiki/Sigma-Discrete_Set_of_Subsets_is_Sigma-Locally_Finite | [
"Sigma-Discrete Sets of Subsets",
"Sigma-Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Sigma-Discrete Set of Subsets",
"Definition:Sigma-Locally Finite Set of Subsets"
] | [
"Discrete Set of Subsets is Locally Finite",
"Category:Sigma-Discrete Sets of Subsets",
"Category:Sigma-Locally Finite Sets of Subsets"
] |
proofwiki-20786 | Recursive Function is Arithmetically Definable | Let $f: \N^k \to \N$ be a recursive function.
Then there exists a $\Sigma_1$ WFF of $k + 1$ free variables:
:$\map \phi {y, x_1, x_2, \dotsc, x_k}$
such that:
:$y = \map f {x_1, x_2, \dotsc, x_k} \iff \N \models \map \phi {\sqbrk y, \sqbrk {x_1}, \sqbrk {x_2}, \dotsc, \sqbrk {x_k} }$
where $\sqbrk a$ denotes the unary ... | By definition of recursive function:
:$f$ can be obtained from basic primitive recursive functions using the operations of:
::substitution
::primitive recursion, and
::minimization on a function
:a finite number of times.
The existence of $\phi$ follows from:
:Basic Primitive Recursive Functions are Arithmetically Def... | Let $f: \N^k \to \N$ be a [[Definition:Recursive Function|recursive function]].
Then there exists a [[Definition:Arithmetical Hierarchy|$\Sigma_1$]] [[Definition:WFF of Predicate Logic|WFF]] of $k + 1$ [[Definition:Free Variable|free variables]]:
:$\map \phi {y, x_1, x_2, \dotsc, x_k}$
such that:
:$y = \map f {x_1, x_... | By definition of [[Definition:Recursive Function|recursive function]]:
:$f$ can be obtained from [[Definition:Basic Primitive Recursive Function|basic primitive recursive functions]] using the operations of:
::[[Definition:Substitution (Mathematical Logic)|substitution]]
::[[Definition:Primitive Recursion|primitive rec... | Recursive Function is Arithmetically Definable | https://proofwiki.org/wiki/Recursive_Function_is_Arithmetically_Definable | https://proofwiki.org/wiki/Recursive_Function_is_Arithmetically_Definable | [
"Recursive Functions"
] | [
"Definition:Recursive/Function",
"Definition:Arithmetical Hierarchy",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Free Variable",
"Unary Representation of Natural Number"
] | [
"Definition:Recursive/Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Primitive Recursion",
"Definition:Minimization/Function",
"Definition:Finite",
"Basic Primitive Recursive Functions are Arithmetically Definable",
"Substitution... |
proofwiki-20787 | Refinement of a Refinement is Refinement of Cover | Let $S$ be a set.
Let $\UU = \set {U_\alpha}$, $\VV = \set {V_\beta}$ and $\WW = \set {W_\gamma}$ be covers of $S$.
Let $\VV$ be a refinement of $\UU$.
Let $\WW$ be a refinement of $\VV$.
Then:
:$\WW$ is a refinement of $\UU$ | Let $W \in \WW$.
By definition of refinement:
:$\exists V \in \VV : W \subseteq V$
Similarly:
:$\exists U \in \UU : V \subseteq U$
From Subset Relation is Transitive:
:$W \subseteq U$
Since $W$ was arbitrary:
:$\forall W \in \WW : \exists U \in \UU : W \subseteq U$
It follows that $\WW$ is a refinement of $\UU$ by defi... | Let $S$ be a [[Definition:Set|set]].
Let $\UU = \set {U_\alpha}$, $\VV = \set {V_\beta}$ and $\WW = \set {W_\gamma}$ be [[Definition:Cover of Set|covers]] of $S$.
Let $\VV$ be a [[Definition:Refinement of Cover|refinement]] of $\UU$.
Let $\WW$ be a [[Definition:Refinement of Cover|refinement]] of $\VV$.
Then:
:$\... | Let $W \in \WW$.
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\exists V \in \VV : W \subseteq V$
Similarly:
:$\exists U \in \UU : V \subseteq U$
From [[Subset Relation is Transitive]]:
:$W \subseteq U$
Since $W$ was arbitrary:
:$\forall W \in \WW : \exists U \in \UU : W \subseteq U$
It foll... | Refinement of a Refinement is Refinement of Cover | https://proofwiki.org/wiki/Refinement_of_a_Refinement_is_Refinement_of_Cover | https://proofwiki.org/wiki/Refinement_of_a_Refinement_is_Refinement_of_Cover | [
"Covers"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Refinement of Cover",
"Definition:Refinement of Cover",
"Definition:Refinement of Cover"
] | [
"Definition:Refinement of Cover",
"Subset Relation is Transitive",
"Definition:Refinement of Cover",
"Category:Covers"
] |
proofwiki-20788 | Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1 | :$\BB$ is a cover of $X$ | Let $x \in X$.
By definition of cover of set:
:$\exists S \in \SS : x \in S$
By definition of union:
:$\exists n \in \N : S \in \SS_n$
From Set is Subset of Union:
:$S \subseteq B_n$
By definition of subset:
:$x \in B_n$
It follows by definition, $\BB$ is a cover of $X$.
{{qed}}
Category:Sigma-Locally Finite Cover has ... | :$\BB$ is a [[Definition:Cover of Set|cover]] of $X$ | Let $x \in X$.
By definition of [[Definition:Cover of Set|cover of set]]:
:$\exists S \in \SS : x \in S$
By definition of [[Definition:Set Union|union]]:
:$\exists n \in \N : S \in \SS_n$
From [[Set is Subset of Union]]:
:$S \subseteq B_n$
By definition of [[Definition:Subset|subset]]:
:$x \in B_n$
It follows by d... | Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1 | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement/Lemma_1 | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement/Lemma_1 | [
"Sigma-Locally Finite Cover has Locally Finite Refinement"
] | [
"Definition:Cover of Set"
] | [
"Definition:Cover of Set",
"Definition:Set Union",
"Set is Subset of Union",
"Definition:Subset",
"Definition:Cover of Set",
"Category:Sigma-Locally Finite Cover has Locally Finite Refinement"
] |
proofwiki-20789 | Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 2 | :$\CC$ is a locally finite refinement of $\BB$ | ==== Lemma 1 ====
{{:Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1}}{{qed|lemma}} | :$\CC$ is a [[Definition:Locally Finite Set of Subsets|locally finite]] [[Definition:Refinement of Cover|refinement]] of $\BB$ | ==== [[Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1|Lemma 1]] ====
{{:Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1}}{{qed|lemma}} | Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 2 | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement/Lemma_2 | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement/Lemma_2 | [
"Sigma-Locally Finite Cover has Locally Finite Refinement"
] | [
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1"
] |
proofwiki-20790 | Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement | Let $T = \struct {X, \tau}$ be a topological space.
Let:
:$\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$
where each $\SS_n$ is locally finite for all $n \in \N$.
Let $\CC = \set {C_n : n \in \N}$ be a countable locally finite cover of $X$.
Then:
:there exists a common locally... | Let:
:$\AA = \set {C_n \cap S : n \in \N, S \in \SS_n}$ | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let:
:$\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:Cover of Set|cover]] of $X$
where each $\SS_n$ is [[Definition:Locally Finite Set of ... | Let:
:$\AA = \set {C_n \cap S : n \in \N, S \in \SS_n}$ | Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_and_Countable_Locally_Finite_Cover_have_Common_Locally_Finite_Refinement | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_and_Countable_Locally_Finite_Cover_have_Common_Locally_Finite_Refinement | [
"Covers"
] | [
"Definition:Topological Space",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Cover of Set",
"Definition:Locally Finite Set of Subsets",
"Definition:Countable Set",
"Definition:Locally Finite Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [] |
proofwiki-20791 | Sigma-Locally Finite Cover has Locally Finite Refinement | Let $T = \struct {X, \tau}$ be a topological space.
Let $\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$, where each $\SS_n$ is locally finite for all $n \in \N$.
Then:
:there exists a locally finite refinement $\AA$ of $\SS$. | For each $n \in \N$, let:
:$B_n = \bigcup \SS_n$
Let:
:$\BB = \set {B_n : n \in \N}$ | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:Cover of Set|cover]] of $X$, where each $\SS_n$ is [[Definition:Locally Finite Set of ... | For each $n \in \N$, let:
:$B_n = \bigcup \SS_n$
Let:
:$\BB = \set {B_n : n \in \N}$ | Sigma-Locally Finite Cover has Locally Finite Refinement | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement | https://proofwiki.org/wiki/Sigma-Locally_Finite_Cover_has_Locally_Finite_Refinement | [
"Covers",
"Sigma-Locally Finite Cover has Locally Finite Refinement"
] | [
"Definition:Topological Space",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Cover of Set",
"Definition:Locally Finite Set of Subsets",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [] |
proofwiki-20792 | Nonexistence of Continuous Linear Transformations over Finite Dimensional Vector Space whose Commutator equals Identity | Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be the space of continuous linear transformations.
Let $A, B \in \map {CL} {X, X}$ be continuous linear transformations.
Let $I$ be the identity operator.
Then there is no $A, B$ such that $A \circ B - B \circ A = I$ | {{AimForCont}} there is $A, B$ such that $A \circ B - B \circ A = I$. | Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be the [[Definition:Continuous Linear Transformation Space|space of continuous linear transformations]].
Let $A, B \in \map {CL} {X, X}$ be [[Definition:Continuous Mapping (Normed ... | {{AimForCont}} there is $A, B$ such that $A \circ B - B \circ A = I$. | Nonexistence of Continuous Linear Transformations over Finite Dimensional Vector Space whose Commutator equals Identity | https://proofwiki.org/wiki/Nonexistence_of_Continuous_Linear_Transformations_over_Finite_Dimensional_Vector_Space_whose_Commutator_equals_Identity | https://proofwiki.org/wiki/Nonexistence_of_Continuous_Linear_Transformations_over_Finite_Dimensional_Vector_Space_whose_Commutator_equals_Identity | [
"Continuous Linear Transformations",
"Commutativity"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Linear Transformation/Vector Space",
"Definition:Identity Mapping"
] | [] |
proofwiki-20793 | Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets | Let $T = \struct {S, \tau}$ be a topological space.
Let $\AA$ be a locally finite set of subsets.
Then:
:$\AA$ is a $\sigma$-locally finite set of subsets. | For each $n \in \N$, let:
:$\AA_n = \AA$
Then:
:$\AA = \ds \bigcup_{n \mathop \in \N} \AA_n$
The result follows.
{{qed}}
Category:Locally Finite Sets of Subsets
Category:Sigma-Locally Finite Sets of Subsets
lrpmdhh948t6ste5rn1zsauijnk3xa9 | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\AA$ be a [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]].
Then:
:$\AA$ is a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite set of subsets]]. | For each $n \in \N$, let:
:$\AA_n = \AA$
Then:
:$\AA = \ds \bigcup_{n \mathop \in \N} \AA_n$
The result follows.
{{qed}}
[[Category:Locally Finite Sets of Subsets]]
[[Category:Sigma-Locally Finite Sets of Subsets]]
lrpmdhh948t6ste5rn1zsauijnk3xa9 | Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets | https://proofwiki.org/wiki/Locally_Finite_Set_of_Subsets_is_Sigma-Locally_Finite_Set_of_Subsets | https://proofwiki.org/wiki/Locally_Finite_Set_of_Subsets_is_Sigma-Locally_Finite_Set_of_Subsets | [
"Locally Finite Sets of Subsets",
"Sigma-Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Locally Finite Set of Subsets",
"Definition:Sigma-Locally Finite Set of Subsets"
] | [
"Category:Locally Finite Sets of Subsets",
"Category:Sigma-Locally Finite Sets of Subsets"
] |
proofwiki-20794 | Inverse Mapping on Group of Units in Unital Banach Algebra is Continuous | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.
Let $\map G A$ be the group of units of $A$.
Define $\phi : \map G A \to \map G A$ by:
:$\map \phi x = x^{-1}$
for each $x \in \map G A$.
Then $\phi$ is continuous. | Let $x \in \map G A$ and $y \in A$ be such that:
:$\ds \norm {x - y} < \frac 1 {\norm {x^{-1} } }$
As shown in Group of Units in Unital Banach Algebra is Open, we have $y \in \map G A$ and:
:$\norm {1 - x^{-1} y} < 1$
Then, from Element of Unital Banach Algebra Close to Identity is Invertible, we have:
:$\ds \norm {... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]].
Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Define $\phi : \map G A \to \map G A$ by:
:$\map \phi x = x^{-1}$
for each $x \in \map G A$.
Then $\phi$ is [[Definition:Continuous ... | Let $x \in \map G A$ and $y \in A$ be such that:
:$\ds \norm {x - y} < \frac 1 {\norm {x^{-1} } }$
As shown in [[Group of Units in Unital Banach Algebra is Open]], we have $y \in \map G A$ and:
:$\norm {1 - x^{-1} y} < 1$
Then, from [[Element of Unital Banach Algebra Close to Identity is Invertible]], we have:
... | Inverse Mapping on Group of Units in Unital Banach Algebra is Continuous | https://proofwiki.org/wiki/Inverse_Mapping_on_Group_of_Units_in_Unital_Banach_Algebra_is_Continuous | https://proofwiki.org/wiki/Inverse_Mapping_on_Group_of_Units_in_Unital_Banach_Algebra_is_Continuous | [
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Group of Units",
"Definition:Continuous Function"
] | [
"Group of Units in Unital Banach Algebra is Open",
"Element of Unital Banach Algebra Close to Identity is Invertible",
"Definition:Algebra Norm",
"Definition:Continuous Function",
"Category:Unital Banach Algebras"
] |
proofwiki-20795 | Existence of Transformations whose Commutator equals Identity | Let $\map {C^\infty} \R$ be the space of smooth real functions.
Let $\circ$ denote the composition of mappings.
Let $A : \map {C^\infty} \R \to \map {C^\infty} \R$ be the mapping such that:
:$\forall \phi \in \map {C^\infty} \R : \forall x \in \R : \map {\paren {A \circ \phi} } x := \map {\dfrac {\d \phi} {\d x} } x$
L... | Let $\Psi \in \map {C^\infty} \R$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {A \circ B - B \circ A} \circ \Psi
| r = \map {\paren {A \circ B} } \Psi - \map {\paren {B \circ A} } \Psi
| c = Derivative Operator is Linear Mapping
}}
{{eqn | r = \map A {\map B \Psi} - \map B {\map A \Psi}
| c = {{Defof|Comp... | Let $\map {C^\infty} \R$ be the [[Definition:Space of Smooth Real Functions|space of smooth real functions]].
Let $\circ$ denote the [[Definition:Composition of Mappings|composition of mappings]].
Let $A : \map {C^\infty} \R \to \map {C^\infty} \R$ be the [[Definition:Mapping|mapping]] such that:
:$\forall \phi \in ... | Let $\Psi \in \map {C^\infty} \R$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {A \circ B - B \circ A} \circ \Psi
| r = \map {\paren {A \circ B} } \Psi - \map {\paren {B \circ A} } \Psi
| c = [[Derivative Operator is Linear Mapping]]
}}
{{eqn | r = \map A {\map B \Psi} - \map B {\map A \Psi}
| c = {{Defo... | Existence of Transformations whose Commutator equals Identity | https://proofwiki.org/wiki/Existence_of_Transformations_whose_Commutator_equals_Identity | https://proofwiki.org/wiki/Existence_of_Transformations_whose_Commutator_equals_Identity | [
"Smooth Real Functions",
"Commutativity"
] | [
"Definition:Space of Smooth Real Functions",
"Definition:Composition of Mappings",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Identity Mapping",
"Definition:Abuse of Notation"
] | [
"Derivative Operator is Linear Mapping",
"Product Rule for Derivatives"
] |
proofwiki-20796 | Characterization of Paracompactness in T3 Space/Lemma 4 | :$\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite | Let $A \in \AA$.
By definition of refinement:
:$\exists W \in \WW : A \subseteq W$
From Subsets of Disjoint Sets are Disjoint:
:$\forall V \in \VV : V \cap A \ne \O \leadsto V \cap W \ne \O$
Hence:
:$\set{V \in \VV : V \cap A \ne \O} \subseteq \set{V \in \VV : V \cap W \ne \O}$
We have {{hypothesis}}:
:$\set{V \in \VV ... | :$\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is [[Definition:Finite Set|finite]] | Let $A \in \AA$.
By definition of [[Definition:Refinement of Cover|refinement]]:
:$\exists W \in \WW : A \subseteq W$
From [[Subsets of Disjoint Sets are Disjoint]]:
:$\forall V \in \VV : V \cap A \ne \O \leadsto V \cap W \ne \O$
Hence:
:$\set{V \in \VV : V \cap A \ne \O} \subseteq \set{V \in \VV : V \cap W \ne \... | Characterization of Paracompactness in T3 Space/Lemma 4 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_4 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_4 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Finite Set"
] | [
"Definition:Refinement of Cover",
"Subsets of Disjoint Sets are Disjoint",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Definition:Finite Set",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20797 | Characterization of Paracompactness in T3 Space/Lemma 7 | :$\UU^*$ is an open locally finite refinement of $\UU$ | ==== Lemma 5 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 5}}{{qed|lemma}} | :$\UU^*$ is an [[Definition:Open Locally Finite Set of Subsets|open locally finite]] [[Definition:Refinement|refinement]] of $\UU$ | ==== [[Characterization of Paracompactness in T3 Space/Lemma 5|Lemma 5]] ====
{{:Characterization of Paracompactness in T3 Space/Lemma 5}}{{qed|lemma}} | Characterization of Paracompactness in T3 Space/Lemma 7 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_7 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_7 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Open Locally Finite Set of Subsets",
"Definition:Refinement"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 5"
] |
proofwiki-20798 | Norm of Inverse of Sequence of Invertible Elements Converging to Non-Invertible Element in Unital Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.
Let $\map G A$ be the group of units of $A$.
Let $x \in A \setminus \map G A$.
Let $\sequence {x_n}_{n \in \N}$ be a sequence in $\map G A$ such that $x_n \to x$.
Then $\norm {x_n^{-1} } \to \infty$ as $n \to \infty$. | Note that if there existed $n \in \N$ such that:
:$\ds \norm {x - x_n} < \frac 1 {\norm {x_n^{-1} } }$
then we would have $x \in \map G A$ from Group of Units in Unital Banach Algebra is Open.
So, we have:
:$\ds \frac 1 {\norm {x_n^{-1} } } \le \norm {x - x_n}$ for each $n \in \N$.
Since we have $x_n \to x$, we hav... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]].
Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Let $x \in A \setminus \map G A$.
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Sequence|sequence]] in $\map G A$ such that $x_n \t... | Note that if there existed $n \in \N$ such that:
:$\ds \norm {x - x_n} < \frac 1 {\norm {x_n^{-1} } }$
then we would have $x \in \map G A$ from [[Group of Units in Unital Banach Algebra is Open]].
So, we have:
:$\ds \frac 1 {\norm {x_n^{-1} } } \le \norm {x - x_n}$ for each $n \in \N$.
Since we have $x_n \to x... | Norm of Inverse of Sequence of Invertible Elements Converging to Non-Invertible Element in Unital Banach Algebra | https://proofwiki.org/wiki/Norm_of_Inverse_of_Sequence_of_Invertible_Elements_Converging_to_Non-Invertible_Element_in_Unital_Banach_Algebra | https://proofwiki.org/wiki/Norm_of_Inverse_of_Sequence_of_Invertible_Elements_Converging_to_Non-Invertible_Element_in_Unital_Banach_Algebra | [
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Group of Units",
"Definition:Sequence"
] | [
"Group of Units in Unital Banach Algebra is Open",
"Category:Unital Banach Algebras"
] |
proofwiki-20799 | Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.
Let $\map G A$ be the group of units of $A$.
Let $B$ be a closed subalgebra of $A$.
Let $\map G B$ be the group of units of $A$.
Let $x \in \partial \map G B$, where $\partial \map G B$ is the topological boundary of $\map G B$.
Then $x$ is not inverti... | === Lemma ===
{{:Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma}}{{qed|lemma}}
{{AimForCont}} that there exists $y \in A$ such that $x y = 1$.
From the lemma, there exists a sequence $\sequence {z_n}_{n \in \N}$ in $A$ such that $\norm {z_n} = 1$ for each... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]].
Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Let $B$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Subalgebra|subalgebra]] of $A$.
Let $\map G B$ be the [[Definition:... | === [[Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma|Lemma]] ===
{{:Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma}}{{qed|lemma}}
{{AimForCont}} that there exists $y \in A$ such that $x y ... | Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra | https://proofwiki.org/wiki/Element_of_Unital_Banach_Algebra_on_Boundary_of_Group_of_Units_of_Subalgebra_is_Not_Invertible_in_Algebra | https://proofwiki.org/wiki/Element_of_Unital_Banach_Algebra_on_Boundary_of_Group_of_Units_of_Subalgebra_is_Not_Invertible_in_Algebra | [
"Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra",
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Group of Units",
"Definition:Closed Set/Topology",
"Definition:Subalgebra",
"Definition:Group of Units",
"Definition:Boundary (Topology)",
"Definition:Invertible Element"
] | [
"Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma",
"Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma",
"Definition:Sequence",
"Product Rule for Sequence in Normed Algebra",
"Modulus... |
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