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-16400 | Volume of Solid of Revolution/Parametric Form | Let $x: \R \to \R$ and $y: \R \to \R$ be real functions defined on the interval $\closedint a b$.
Let $y$ be integrable on the (closed) interval $\closedint a b$.
Let $x$ be differentiable on the (open) interval $\openint a b$.
Let the points be defined:
:$A = \tuple {\map x a, \map y a}$
:$B = \tuple {\map x b, \map y... | {{begin-eqn}}
{{eqn|c = Volume of Solid of Revolution
|l = V
|r = \pi \int_a^b \paren {\map f x}^2 \rd x
}}
{{eqn|c = Integration by Substitution/Definite Integral
|r = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t
}}
{{end-eqn}}
{{qed}} | Let $x: \R \to \R$ and $y: \R \to \R$ be [[Definition:Real Function|real functions]] defined on the [[Definition:Closed Real Interval|interval]] $\closedint a b$.
Let $y$ be [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|(closed) interval]] $\closedint a b$.
Let $x$ be [[Defini... | {{begin-eqn}}
{{eqn|c = [[Volume of Solid of Revolution]]
|l = V
|r = \pi \int_a^b \paren {\map f x}^2 \rd x
}}
{{eqn|c = [[Integration by Substitution/Definite Integral]]
|r = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t
}}
{{end-eqn}}
{{qed}} | Volume of Solid of Revolution/Parametric Form | https://proofwiki.org/wiki/Volume_of_Solid_of_Revolution/Parametric_Form | https://proofwiki.org/wiki/Volume_of_Solid_of_Revolution/Parametric_Form | [
"Integral Calculus",
"Solids of Revolution"
] | [
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Differentiable",
"Definition:Real Interval/Open",
"Definition:Point",
"Definition:Geometric Figure/Plane Figure",
"Definition:Line/Straight Line",
"Defini... | [
"Volume of Solid of Revolution",
"Integration by Substitution/Definite Integral"
] |
proofwiki-16401 | Weight of Body at Earth's Surface | Let $B$ be a body of mass $m$ situated at (or near) the surface of Earth.
Then the weight of $B$ is given by:
:$W = m g$
where $g$ is the value of the acceleration due to gravity at the surface of Earth. | The weight of $B$ is the magnitude of the force exerted on it by the influence of the gravitational field it is in.
By Newton's Second Law of Motion, that force is given by:
:$\mathbf W = -m g \mathbf k$
where:
:$g$ is the value of the acceleration due to gravity at the surface of Earth
:$\mathbf k$ is a unit vector di... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Mass|mass]] $m$ situated at (or near) the surface of [[Definition:Earth|Earth]].
Then the [[Definition:Weight (Physics)|weight]] of $B$ is given by:
:$W = m g$
where $g$ is the value of the [[Acceleration Due to Gravity|acceleration due to gravity]] at the surfa... | The [[Definition:Weight (Physics)|weight]] of $B$ is the [[Definition:Magnitude|magnitude]] of the [[Definition:Force|force]] exerted on it by the influence of the [[Definition:Gravitational Field|gravitational field]] it is in.
By [[Newton's Second Law of Motion]], that [[Definition:Force|force]] is given by:
:$\mat... | Weight of Body at Earth's Surface | https://proofwiki.org/wiki/Weight_of_Body_at_Earth's_Surface | https://proofwiki.org/wiki/Weight_of_Body_at_Earth's_Surface | [
"Weight (Physics)"
] | [
"Definition:Body",
"Definition:Mass",
"Definition:Earth",
"Definition:Weight (Physics)",
"Acceleration Due to Gravity",
"Definition:Earth"
] | [
"Definition:Weight (Physics)",
"Definition:Magnitude",
"Definition:Force",
"Definition:Gravitational Field",
"Newton's Laws of Motion/Second Law",
"Definition:Force",
"Acceleration Due to Gravity",
"Definition:Earth",
"Definition:Unit Vector",
"Definition:Vertical",
"Definition:Magnitude"
] |
proofwiki-16402 | Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule | :$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological ring, $\struct {R, +, \tau_{_R} }$ is a topological group.
From Product Rule for Continuous Mappings to Topological Group:
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Ring|topological ring]], $\struct {R, +, \tau_{_R} }$ is a [[Definition:Topological Group|topological group]].
From [[Combination Theorem for Continuous Mappings/Topological Group/Product Rule|Product Rule for Continuous Mappings to Topological Group]]:
:$f + g: \struct {S, ... | Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Sum_Rule | [
"Combination Theorem for Continuous Mappings to Topological Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Ring",
"Definition:Topological Group",
"Combination Theorem for Continuous Mappings/Topological Group/Product Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16403 | Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule | :$-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological ring:
:$\struct {R, +, \tau_{_R} }$ is a topological group.
From Inverse Rule for Continuous Mappings to Topological Group:
:$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Ring|topological ring]]:
:$\struct {R, +, \tau_{_R} }$ is a [[Definition:Topological Group|topological group]].
From [[Inverse Rule for Continuous Mappings to Topological Group]]:
:$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:Continuous Mapping... | Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Negation_Rule | [
"Combination Theorem for Continuous Mappings to Topological Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Ring",
"Definition:Topological Group",
"Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16404 | Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological ring:
:$\struct {R, *, \tau_{_R} }$ is a topological semigroup.
From Multiple Rule for Continuous Mappings to Topological Semigroup:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ are continuous mappings.
{{qed}} | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Ring|topological ring]]:
:$\struct {R, *, \tau_{_R} }$ is a [[Definition:Topological Semigroup|topological semigroup]].
From [[Multiple Rule for Continuous Mappings to Topological Semigroup]]:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ a... | Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Multiple_Rule | [
"Combination Theorem for Continuous Mappings to Topological Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Ring",
"Definition:Topological Semigroup",
"Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16405 | Combination Theorem for Continuous Mappings/Topological Ring/Product Rule | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological ring:
:$\struct {R, *, \tau_{_R} }$ is a topological semigroup.
From Product Rule for Continuous Mappings to Topological Semigroup:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Ring|topological ring]]:
:$\struct {R, *, \tau_{_R} }$ is a [[Definition:Topological Semigroup|topological semigroup]].
From [[Product Rule for Continuous Mappings to Topological Semigroup]]:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:Con... | Combination Theorem for Continuous Mappings/Topological Ring/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Product_Rule | [
"Combination Theorem for Continuous Mappings to Topological Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Ring",
"Definition:Topological Semigroup",
"Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16406 | Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule | :$\lambda * f + \mu * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping
:$f * \lambda + g * \mu: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping. | {{ProofWanted}}
Category:Combination Theorem for Continuous Mappings
mfas2n1k3vr4wtip9erhca8c1e96jr6 | :$\lambda * f + \mu * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]
:$f * \lambda + g * \mu: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | {{ProofWanted}}
[[Category:Combination Theorem for Continuous Mappings]]
mfas2n1k3vr4wtip9erhca8c1e96jr6 | Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Combined_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Combined_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Category:Combination Theorem for Continuous Mappings"
] |
proofwiki-16407 | Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule | :$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological division ring, $\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Sum Rule for Continuous Mappings into Topological Ring:
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Division Ring|topological division ring]], $\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Ring|topological ring]].
From [[Sum Rule for Continuous Mappings into Topological Ring]]:
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:... | Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Sum_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Definition:Topological Ring",
"Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16408 | Combination Theorem for Continuous Mappings/Topological Division Ring/Multiple Rule | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological division ring:
:$\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Multiple Rule for Continuous Mappings to Topological Ring:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ are continuous mappings.
{{qed}} | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Division Ring|topological division ring]]:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Ring|topological ring]].
From [[Multiple Rule for Continuous Mappings to Topological Ring]]:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R... | Combination Theorem for Continuous Mappings/Topological Division Ring/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Multiple_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Definition:Topological Ring",
"Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16409 | Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological division ring:
:$\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Product Rule for Continuous Mappings to Topological Ring:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Division Ring|topological division ring]]:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Ring|topological ring]].
From [[Product Rule for Continuous Mappings to Topological Ring]]:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definiti... | Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Product_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Definition:Topological Ring",
"Combination Theorem for Continuous Mappings/Topological Ring/Product Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16410 | Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule | :$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous. | Let $R^* = R \setminus \set 0$.
Let $\tau^*$ be the subspace topology on $R^*$.
By definition of a topological division ring:
:$\phi: \struct {R^*, \tau^*} \to \struct {R, \tau_{_R} }$ such that $\forall x \in R^*: \map \phi x = x^{-1}$ is a continuous mapping
Let $g^*: \struct {U, \tau_{_U} } \to \struct {R^*, \tau^*}... | :$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | Let $R^* = R \setminus \set 0$.
Let $\tau^*$ be the [[Definition:Subspace Topology|subspace topology]] on $R^*$.
By definition of a [[Definition:Topological Division Ring|topological division ring]]:
:$\phi: \struct {R^*, \tau^*} \to \struct {R, \tau_{_R} }$ such that $\forall x \in R^*: \map \phi x = x^{-1}$ is a [[... | Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Inverse_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Inverse_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Subspace",
"Definition:Topological Division Ring",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition: Restriction of Mapping",
"Restriction of Continuous Mapping is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Composite of Continuous Mappi... |
proofwiki-16411 | Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule | :$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological division ring:
:$\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Negation Rule for Continuous Mappings to Topological Ring:
:$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Division Ring|topological division ring]]:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Ring|topological ring]].
From [[Negation Rule for Continuous Mappings to Topological Ring]]:
:$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition... | Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Negation_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Definition:Topological Ring",
"Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16412 | Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological ring:
:$\struct {R, +, \tau_{_R} }$ is a topological group.
From Multiple Rule for Continuous Mappings to Topological Group:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Ring|topological ring]]:
:$\struct {R, +, \tau_{_R} }$ is a [[Definition:Topological Group|topological group]].
From [[Multiple Rule for Continuous Mappings to Topological Group]]:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Definition:Continuo... | Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Translation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Ring/Translation_Rule | [
"Combination Theorem for Continuous Mappings to Topological Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Ring",
"Definition:Topological Group",
"Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16413 | Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | By definition of a topological division ring:
:$\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Translation Rule for Continuous Mappings to Topological Ring:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | By definition of a [[Definition:Topological Division Ring|topological division ring]]:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Ring|topological ring]].
From [[Translation Rule for Continuous Mappings to Topological Ring]]:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a ... | Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Translation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Division_Ring/Translation_Rule | [
"Combination Theorem for Continuous Mappings to Topological Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Definition:Topological Ring",
"Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16414 | Pointwise Operation is Composite of Operation with Mapping to Cartesian Product | Let $S$ be a set.
Let $\struct {T, *}$ be an algebraic structure.
Let $T^S$ be the set of all mappings from $S$ to $T$.
Let the algebraic structure $\struct {T^S, \oplus}$ be the algebraic structure on $T^S$ induced by $*$.
Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.
Let $f \times g : S ... | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {* \circ \paren {f \times g} } } x
| r = \map * {\map {\paren {f \times g} } x}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map * {\tuple {\map f x, \map g x} }
| c = Definition of $f \times g$
}}
{{eqn | r = \map f x * \ma... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {T, *}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $T^S$ be the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to $T$.
Let the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struc... | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {* \circ \paren {f \times g} } } x
| r = \map * {\map {\paren {f \times g} } x}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map * {\tuple {\map f x, \map g x} }
| c = Definition of $f \times g$
}}
{{eqn | r = \map f x * \ma... | Pointwise Operation is Composite of Operation with Mapping to Cartesian Product | https://proofwiki.org/wiki/Pointwise_Operation_is_Composite_of_Operation_with_Mapping_to_Cartesian_Product | https://proofwiki.org/wiki/Pointwise_Operation_is_Composite_of_Operation_with_Mapping_to_Cartesian_Product | [
"Abstract Algebra",
"Pointwise Operations"
] | [
"Definition:Set",
"Definition:Algebraic Structure/One Operation",
"Definition:Set of All Mappings",
"Definition:Algebraic Structure/One Operation",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Cartesian Product",
"Definition:Composition ... | [
"Equality of Mappings",
"Category:Abstract Algebra",
"Category:Pointwise Operations"
] |
proofwiki-16415 | Integral Representation of Bessel Function of the First Kind/Integer Order | Let $n \in \Z$ be an integer.
Then:
:$\ds \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$ | {{begin-eqn}}
{{eqn | l = \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} }
| r = \sum_{m \mathop = -\infty}^\infty \map {J_m} x t^m
| c = Generating Function for Bessel Function of the First Kind of Order n of x
}}
{{eqn | l = \dfrac 1 {2 \pi i} \int_C t^{-n - 1} \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t}... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\ds \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$ | {{begin-eqn}}
{{eqn | l = \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} }
| r = \sum_{m \mathop = -\infty}^\infty \map {J_m} x t^m
| c = [[Generating Function for Bessel Function of the First Kind of Order n of x]]
}}
{{eqn | l = \dfrac 1 {2 \pi i} \int_C t^{-n - 1} \map \exp {\dfrac x 2 \paren {t - \dfrac ... | Integral Representation of Bessel Function of the First Kind/Integer Order | https://proofwiki.org/wiki/Integral_Representation_of_Bessel_Function_of_the_First_Kind/Integer_Order | https://proofwiki.org/wiki/Integral_Representation_of_Bessel_Function_of_the_First_Kind/Integer_Order | [
"Bessel Functions"
] | [
"Definition:Integer"
] | [
"Generating Function for Bessel Function of the First Kind of Order n of x",
"Definition:Contour Integral/Complex",
"Definition:Coordinate System/Origin",
"Cauchy's Residue Theorem",
"Definition:Unit Circle",
"Definition:Anticlockwise",
"Sine in terms of Hyperbolic Sine",
"Cosine in terms of Hyperboli... |
proofwiki-16416 | Integral Representation of Bessel Function of the First Kind/Non-Integer Order | Let $n \in \Z$ be an integer.
Then:
:$\ds \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$ | {{ProofWanted|big job}} | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\ds \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$ | {{ProofWanted|big job}} | Integral Representation of Bessel Function of the First Kind/Non-Integer Order | https://proofwiki.org/wiki/Integral_Representation_of_Bessel_Function_of_the_First_Kind/Non-Integer_Order | https://proofwiki.org/wiki/Integral_Representation_of_Bessel_Function_of_the_First_Kind/Non-Integer_Order | [
"Bessel Functions"
] | [
"Definition:Integer"
] | [] |
proofwiki-16417 | Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct{R, +, *, \tau_{_R} }$ is a topological division ring.
From Translation Rule for Continuous Mappings to Topological Division Ring:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct{R, +, *, \tau_{_R} }$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Translation Rule for Continuous Mappings to Topological Division Ring]]:
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$... | Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Translation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Translation_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16418 | Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule | :$- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}, $\struct{R, +, *, \tau_{_R}}$ is a topological division ring.
From Negation Rule for Continuous Mappings to Topological Division Ring, $- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.
{{qed}} | :$- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}, $\struct{R, +, *, \tau_{_R}}$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule|Negation Rule for Continuous Mappings to Topological Divis... | Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Negation_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Negation_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16419 | Legendre Transform of Strictly Convex Real Function is Strictly Convex | Let $\map f x$ be a strictly convex real function.
Then the function $\map {f^*} p$ acquired through the Legendre Transform is also strictly convex. | {{begin-eqn}}
{{eqn | l = \frac {\d f^*} {\d p}
| r = -\frac {\d \map f {\map x p} } {\d p} + \frac {\map \d {p \map x p} } {\d p}
| c = {{Defof|Legendre Transform}}
}}
{{eqn | r = -f' \frac {\d x} {\d p} + x + p \frac {\d x} {\d p}
| c = Product Rule for Derivatives
}}
{{eqn | r = -p \frac {\d x} {\d... | Let $\map f x$ be a [[Definition:Convex Real Function/Definition 1/Strictly|strictly convex]] [[Definition:Real Function|real function]].
Then the [[Definition:Real Function|function]] $\map {f^*} p$ acquired through [[Definition:Legendre Transform|the Legendre Transform]] is also [[Definition:Convex Real Function/De... | {{begin-eqn}}
{{eqn | l = \frac {\d f^*} {\d p}
| r = -\frac {\d \map f {\map x p} } {\d p} + \frac {\map \d {p \map x p} } {\d p}
| c = {{Defof|Legendre Transform}}
}}
{{eqn | r = -f' \frac {\d x} {\d p} + x + p \frac {\d x} {\d p}
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = -p \frac {\d x}... | Legendre Transform of Strictly Convex Real Function is Strictly Convex | https://proofwiki.org/wiki/Legendre_Transform_of_Strictly_Convex_Real_Function_is_Strictly_Convex | https://proofwiki.org/wiki/Legendre_Transform_of_Strictly_Convex_Real_Function_is_Strictly_Convex | [
"Calculus of Variations"
] | [
"Definition:Convex Real Function/Definition 1/Strictly",
"Definition:Real Function",
"Definition:Real Function",
"Definition:Legendre Transform",
"Definition:Convex Real Function/Definition 1/Strictly"
] | [
"Product Rule for Derivatives",
"Definition:Legendre Transform",
"Derivative of Inverse Function",
"Definition:Legendre Transform",
"Definition:Real Function",
"Definition:Strictly Convex Real Function",
"Real Function is Strictly Convex iff Derivative is Strictly Increasing",
"Definition:Strictly Inc... |
proofwiki-16420 | Additive Regular Representations of Topological Ring are Homeomorphisms | Let $\struct {R, + , \circ, \tau}$ be a topological ring.
Let $x \in R$.
Let $\lambda_x$ and $\rho_x$ be the left and right regular representations of $\struct {R, +}$ with respect to $x$.
Then $\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\lambda_{-x}, \,\rho_{-x}: \s... | By definition of a topological ring, $ \struct {R, + , \tau}$ is a topological group.
From Right and Left Regular Representations in Topological Group are Homeomorphisms:
:$\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\,\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to ... | Let $\struct {R, + , \circ, \tau}$ be a [[Definition:Topological Ring|topological ring]].
Let $x \in R$.
Let $\lambda_x$ and $\rho_x$ be the [[Definition:Left Regular Representation|left]] and [[Definition:Right Regular Representation|right]] [[Definition:Regular Representations|regular representations]] of $\struct ... | By definition of a [[Definition:Topological Ring|topological ring]], $ \struct {R, + , \tau}$ is a [[Definition:Topological Group|topological group]].
From [[Right and Left Regular Representations in Topological Group are Homeomorphisms]]:
:$\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are [[Definitio... | Additive Regular Representations of Topological Ring are Homeomorphisms | https://proofwiki.org/wiki/Additive_Regular_Representations_of_Topological_Ring_are_Homeomorphisms | https://proofwiki.org/wiki/Additive_Regular_Representations_of_Topological_Ring_are_Homeomorphisms | [
"Topological Rings"
] | [
"Definition:Topological Ring",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Regular Representations",
"Definition:Homeomorphism",
"Definition:Inverse Mapping"
] | [
"Definition:Topological Ring",
"Definition:Topological Group",
"Right and Left Regular Representations in Topological Group are Homeomorphisms",
"Definition:Homeomorphism",
"Definition:Inverse Mapping",
"Category:Topological Rings"
] |
proofwiki-16421 | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms | Let $\struct{R, + , \circ, \tau}$ be a topological ring with unity $1_R$.
For all $y \in R$, let $\lambda_y$ and $\rho_y$ denote the left and right regular representations of $\struct{R, \circ}$ with respect to $y$.
Let $x \in R$ be a unit of $R$ with product inverse $x^{-1}$.
Then $\lambda_x, \, \rho_x: \struct{R, \ta... | Let $I_{_R} : R \to R$ be the identity mapping on $R$.
For all $y \in R$, let $y * I_{_R} : R \to R$ be the mapping defined by:
:$\forall z \in R: \map {\paren {y * I_{_R} } } z = y * \map {I_{_R}} z$
For all $y \in R$, let $I_{_R} * y : R \to R$ be the mapping defined by:
:$\forall z \in R: \map {\paren {I_{_R} * y}} ... | Let $\struct{R, + , \circ, \tau}$ be a [[Definition:Topological Ring|topological ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
For all $y \in R$, let $\lambda_y$ and $\rho_y$ denote the [[Definition:Left Regular Representation|left]] and [[Definition:Right Regular Representation|right]] [[Definition:Regular Re... | Let $I_{_R} : R \to R$ be the [[Definition:Identity Mapping|identity mapping]] on $R$.
For all $y \in R$, let $y * I_{_R} : R \to R$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall z \in R: \map {\paren {y * I_{_R} } } z = y * \map {I_{_R}} z$
For all $y \in R$, let $I_{_R} * y : R \to R$ be the [[Definit... | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms | [
"Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms",
"Topological Rings"
] | [
"Definition:Topological Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Regular Representations",
"Definition:Unit of Ring",
"Definition:Product Inverse",
... | [
"Definition:Identity Mapping",
"Definition:Mapping",
"Definition:Mapping"
] |
proofwiki-16422 | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1 | :$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$ | Let $y \in R$.
{{begin-eqn}}
{{eqn | q = \forall z \in R
| l = \map {\paren {y * I_{_R} } } z
| r = y * \map {I_{_R} } z
| c = Definition of $y * I_{_R}$
}}
{{eqn | r = y * z
| c = {{Defof|Identity Mapping}} $I_{_R}$
}}
{{eqn | r = \map {\lambda_y} z
| c = {{Defof|Left Regular Representati... | :$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$ | Let $y \in R$.
{{begin-eqn}}
{{eqn | q = \forall z \in R
| l = \map {\paren {y * I_{_R} } } z
| r = y * \map {I_{_R} } z
| c = Definition of $y * I_{_R}$
}}
{{eqn | r = y * z
| c = {{Defof|Identity Mapping}} $I_{_R}$
}}
{{eqn | r = \map {\lambda_y} z
| c = {{Defof|Left Regular Representat... | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1 | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms/Lemma_1 | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms/Lemma_1 | [
"Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms"
] | [] | [
"Equality of Mappings",
"Equality of Mappings",
"Category:Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms"
] |
proofwiki-16423 | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2 | :$x * I_R$ is a bijection and $x^{-1} * I_R$ is the inverse of $x * I_R$
:$I_R * x$ is a bijection and $I_R * x^{-1}$ is the inverse of $I_R * x$ | Consider the composite of $x * I_R$ with $x^{-1} * I_R$.
{{begin-eqn}}
{{eqn | q = \forall y \in R
| l = \map {\paren {\paren {x * I_R} \circ \paren {x^{-1} * I_R} } } y
| r = \map {\paren {x * I_R} } {\map {\paren {x^{-1} * I_R} } y}
| c =
}}
{{eqn | r = \map {\paren {x * I_R} } {x^{-1} * y}
|... | :$x * I_R$ is a [[Definition:Bijection|bijection]] and $x^{-1} * I_R$ is the [[Definition:Inverse Mapping|inverse]] of $x * I_R$
:$I_R * x$ is a [[Definition:Bijection|bijection]] and $I_R * x^{-1}$ is the [[Definition:Inverse Mapping|inverse]] of $I_R * x$ | Consider the [[Definition:Composite Mapping|composite]] of $x * I_R$ with $x^{-1} * I_R$.
{{begin-eqn}}
{{eqn | q = \forall y \in R
| l = \map {\paren {\paren {x * I_R} \circ \paren {x^{-1} * I_R} } } y
| r = \map {\paren {x * I_R} } {\map {\paren {x^{-1} * I_R} } y}
| c =
}}
{{eqn | r = \map {\paren... | Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2 | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms/Lemma_2 | https://proofwiki.org/wiki/Multiplicative_Regular_Representations_of_Units_of_Topological_Ring_are_Homeomorphisms/Lemma_2 | [
"Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms"
] | [
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Inverse Mapping"
] | [
"Definition:Composition of Mappings",
"Equality of Mappings",
"Definition:Composition of Mappings",
"Equality of Mappings",
"Definition:Left Inverse Mapping",
"Definition:Right Inverse Mapping",
"Definition:Bijection/Definition 2",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:... |
proofwiki-16424 | Open Balls of P-adic Number | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.
Then:
:$\forall n \in Z : \map {B_{p^{-n} } } a = a + p^{n + 1... | Let $n \in \Z$.
From Open Ball in P-adic Numbers is Closed Ball:
:$\map {B_{p^{-n} } } a = \map {B^{\,-}_{p^{-\paren {n + 1} } } } a$
From Closed Balls of P-adic Number:
:$\map { B^{\,-}_{p^{-\paren {n + 1} } } } a = a + p^{n + 1} \Z_p$
The result follows.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denot... | Let $n \in \Z$.
From [[Open Ball in P-adic Numbers is Closed Ball]]:
:$\map {B_{p^{-n} } } a = \map {B^{\,-}_{p^{-\paren {n + 1} } } } a$
From [[Closed Balls of P-adic Number]]:
:$\map { B^{\,-}_{p^{-\paren {n + 1} } } } a = a + p^{n + 1} \Z_p$
The result follows.
{{qed}} | Open Balls of P-adic Number | https://proofwiki.org/wiki/Open_Balls_of_P-adic_Number | https://proofwiki.org/wiki/Open_Balls_of_P-adic_Number | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Open Ball/Normed Division Ring",
"Definition:Open Ball/Normed Division Ring/Radius"
] | [
"Open Ball in P-adic Numbers is Closed Ball",
"Closed Ball of P-adic Number"
] |
proofwiki-16425 | Local Basis of P-adic Number | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Then the set of open balls $\set {\map {B_{p^{-n} } } a : n \in Z}$ is a local basis of $a$ consisting of clopen sets. | Let $\BB_a$ be the set of all open balls of $Q_p$ centered on $a$.
That is:
:$\BB_a = \set{\map {B_\epsilon} a : \epsilon \in \R_{>0}}$
From Open Balls Centered on P-adic Number is Countable:
:$\BB_a = \set {\map {B_{p^{-n} } } a : n \in Z}$
From Open Balls form Local Basis for Point of Metric Space, $\BB_a$ is a local... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Then the [[Definition:Set|set]] of [[Definition:Open Ball in P-adic Numbers|open balls]] $\set {\map {B_{p^{-n} } } a : n \in Z}$ ... | Let $\BB_a$ be the [[Definition:Set|set]] of all [[Definition:Open Ball|open balls]] of $Q_p$ [[Definition:Center of Open Ball in P-adic Numbers|centered]] on $a$.
That is:
:$\BB_a = \set{\map {B_\epsilon} a : \epsilon \in \R_{>0}}$
From [[Open Balls Centered on P-adic Number is Countable]]:
:$\BB_a = \set {\map {B_{... | Local Basis of P-adic Number | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Set",
"Definition:Open Ball/P-adic Numbers",
"Definition:Local Basis",
"Definition:Clopen Set"
] | [
"Definition:Set",
"Definition:Open Ball",
"Definition:Open Ball/P-adic Numbers/Center",
"Closed Balls Centered on P-adic Number is Countable/Open Balls",
"Open Balls form Local Basis for Point of Metric Space",
"Definition:Local Basis",
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
... |
proofwiki-16426 | Closed Ball of P-adic Number | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ denote the $p$-adic integers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.
Then:
:$\forall n \in Z : \map {B^-_{p^{-n} } } a =... | Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B^{\,-}_{p^{-n} } } a
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}_p
| o = \le
| r = p^{-n}
| c = {{Defof|Closed Ball in P-adic Numbers}}
}}
{{eqn | ll= \leadstoandfrom
| l = p^n \norm {x... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ denote the [[Definition:P-adic Integer|$p$-adic integers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-... | Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B^{\,-}_{p^{-n} } } a
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}_p
| o = \le
| r = p^{-n}
| c = {{Defof|Closed Ball in P-adic Numbers}}
}}
{{eqn | ll= \leadstoandfrom
| l = p^n \norm {... | Closed Ball of P-adic Number | https://proofwiki.org/wiki/Closed_Ball_of_P-adic_Number | https://proofwiki.org/wiki/Closed_Ball_of_P-adic_Number | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers/Radius",
"Definition:Coset/Left Coset",
"Definition:Principal Ideal",
"Definition:Subring",
"Definition:Closed Ball/P-a... | [
"Definition:Set Equality"
] |
proofwiki-16427 | Closed Ball is Disjoint Union of Open Balls in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$
and:
:let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.
Then:
:$(1): \quad \forall n \... | Let $n \in \Z$.
From Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers:
:$\map {B^-_{p^{-n} } } a = \ds \bigcup_{i \mathop = 0}^{p - 1} \map {B^-_{p^{-\paren {n + 1} } } } {a + i p}$
where $\set {\map {B^-_{p^{-\paren {n + 1 } } } } {a + i p^n} : i = 0, \dots, p - 1}$ is a set of pairwise disjoint... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:let $\map { {B_\epsilon}^-} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed $\epsi... | Let $n \in \Z$.
From [[Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers]]:
:$\map {B^-_{p^{-n} } } a = \ds \bigcup_{i \mathop = 0}^{p - 1} \map {B^-_{p^{-\paren {n + 1} } } } {a + i p}$
where $\set {\map {B^-_{p^{-\paren {n + 1 } } } } {a + i p^n} : i = 0, \dots, p - 1}$ is a [[Definition:Set|se... | Closed Ball is Disjoint Union of Open Balls in P-adic Numbers | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Open_Balls_in_P-adic_Numbers | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Open_Balls_in_P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Open Ball/P-adic Numbers"
] | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Closed Ball/P-adic Numbers",
"Open Ball in P-adic Numbers is Closed Ball",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Open Ball/P-adic Numbers"
] |
proofwiki-16428 | Sphere is Disjoint Union of Open Balls in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:let $\map {S_\epsilon} a$ denote the sphere of $a$ of radius $\epsilon$.
:let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius... | For all $\epsilon \in \R_{>0}$:
:let $\map {B^-_\epsilon} a$ denote the closed ball of $a$ of radius $\epsilon$.
Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \map {S_{p^{-n} } } a
| r = \map {B^-_{p^{-n} } } a \setminus \map {B_{p^{-n} } } a
| c = Sphere is Set Difference of Closed and Open Ball in P-ad... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:let $\map {S_\epsilon} a$ deno... | For all $\epsilon \in \R_{>0}$:
:let $\map {B^-_\epsilon} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed ball]] of $a$ of [[Definition:Radius of Open Ball in P-adic Numbers|radius]] $\epsilon$.
Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \map {S_{p^{-n} } } a
| r = \map {B^-_{p^{-n} } } a... | Sphere is Disjoint Union of Open Balls in P-adic Numbers | https://proofwiki.org/wiki/Sphere_is_Disjoint_Union_of_Open_Balls_in_P-adic_Numbers | https://proofwiki.org/wiki/Sphere_is_Disjoint_Union_of_Open_Balls_in_P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Sphere/P-adic Numbers",
"Definition:Sphere/P-adic Numbers/Radius",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Radius"
] | [
"Definition:Closed Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Radius",
"Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers",
"Closed Ball is Disjoint Union of Open Balls in P-adic Numbers",
"Union is Associative",
"Union is Commutative",
"Set Difference with Union is S... |
proofwiki-16429 | Null Sequence induces Local Basis in Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\sequence {x_n}$ be a real null sequence such that:
:$\forall n \in N: x_n > 0$
Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ in $M$.
Then:
:$\BB_{\sequence {x_n} } = \set{\map {B_{x_n} } a : n \in \N}$ is a local basis at $a$. | By Open Ball of Metric Space is Open Set, every element of $\BB_{\sequence {x_n} }$ is an open neighborhood of $a$.
Let $U$ be an open neighborhood of $a$.
By definition of an open set, there exists a strictly positive real number $\epsilon$ such that $\map {B_\epsilon} a \subseteq U$.
By definition of a real null sequ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\sequence {x_n}$ be a [[Definition:Real Null Sequence|real null sequence]] such that:
:$\forall n \in N: x_n > 0$
Let $\map {B_\epsilon} a$ denote the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball of $a$ in $M$]... | By [[Open Ball of Metric Space is Open Set]], every [[Definition:Element|element]] of $\BB_{\sequence {x_n} }$ is an [[Definition:Open Neighborhood of Point|open neighborhood]] of $a$.
Let $U$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $a$.
By definition of an [[Definition:Open Set (Metric ... | Null Sequence induces Local Basis in Metric Space | https://proofwiki.org/wiki/Null_Sequence_induces_Local_Basis_in_Metric_Space | https://proofwiki.org/wiki/Null_Sequence_induces_Local_Basis_in_Metric_Space | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Null Sequence/Real Numbers",
"Definition:Open Ball",
"Definition:Local Basis"
] | [
"Open Ball is Open Set/Pseudometric Space",
"Definition:Element",
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Point",
"Definition:Open Set/Metric Space",
"Definition:Strictly Positive/Real Number",
"Definition:Null Sequence/Real Numbers",
"Subset Relation is Transitive",
"Cat... |
proofwiki-16430 | Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals | :$\BB = \set {\map {B_{1/n}} a : n \in \N}$ is a local basis at $a$. | Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
:$x_n = \dfrac 1 n$
From Sequence of Reciprocals is Null Sequence, $\sequence {x_n}$ is a real null sequence.
From Null Sequence induces Local Basis in Metric Space:
:$\BB = \set {\map {B_{1/n} } a : n \in \N}$ is a local basis at $a$.
{{qed}}
Category:Metric Sp... | :$\BB = \set {\map {B_{1/n}} a : n \in \N}$ is a [[Definition:Local Basis|local basis]] at $a$. | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$x_n = \dfrac 1 n$
From [[Sequence of Reciprocals is Null Sequence]], $\sequence {x_n}$ is a [[Definition:Real Null Sequence|real null sequence]].
From [[Null Sequence induces Local Basis in Metric Space]]:
:$\BB = \set {\map {B_{... | Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals | https://proofwiki.org/wiki/Null_Sequence_induces_Local_Basis_in_Metric_Space/Sequence_of_Reciprocals | https://proofwiki.org/wiki/Null_Sequence_induces_Local_Basis_in_Metric_Space/Sequence_of_Reciprocals | [
"Metric Spaces"
] | [
"Definition:Local Basis"
] | [
"Definition:Real Sequence",
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Definition:Null Sequence/Real Numbers",
"Null Sequence induces Local Basis in Metric Space",
"Definition:Local Basis",
"Category:Metric Spaces"
] |
proofwiki-16431 | Consecutive Integers which are Powers of 2 or 3 | The only pairs of consecutive positive integers which are powers of $2$ or $3$ are:
:$\tuple {1, 2}$, $\tuple {2, 3}$, $\tuple {3, 4}$, $\tuple {8, 9}$ | Let $a$ and $b$ be two arbitrary consecutive positive integers. | The only [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Positive Integer|positive integers]] which are [[Definition:Integer Power|powers]] of $2$ or $3$ are:
:$\tuple {1, 2}$, $\tuple {2, 3}$, $\tuple {3, 4}$, $\tuple {8, 9}$ | Let $a$ and $b$ be two arbitrary consecutive [[Definition:Positive Integer|positive integers]]. | Consecutive Integers which are Powers of 2 or 3 | https://proofwiki.org/wiki/Consecutive_Integers_which_are_Powers_of_2_or_3 | https://proofwiki.org/wiki/Consecutive_Integers_which_are_Powers_of_2_or_3 | [
"Powers of 2",
"Powers of 3"
] | [
"Definition:Ordered Pair",
"Definition:Positive/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Positive/Integer",
"Definition:Positive/Integer"
] |
proofwiki-16432 | Kinetic Energy of Classical Particle | Let $\MM$ be an $n$-dimensional Euclidean manifold.
Let $P$ be a particle with an inertial mass $m_i$.
Let $t$ be the time variable of $P$.
Suppose the position of $P$ is a real differentiable $n$-dimensional vector-valued mapping $\mathbf x = \map {\mathbf x} t$.
Then the '''kinetic energy''' of a '''classical particl... | {{ProofWanted|dependent upon the axiomatic framework}} | Let $\MM$ be an $n$-[[Definition:Dimension (Geometry)|dimensional]] [[Definition:Real Euclidean Space|Euclidean]] [[Definition:Riemannian Manifold|manifold]].
Let $P$ be a [[Definition:Particle|particle]] with an [[Definition:Inertial Mass|inertial mass]] $m_i$.
Let $t$ be the [[Definition:Time|time]] [[Definition:Re... | {{ProofWanted|dependent upon the axiomatic framework}} | Kinetic Energy of Classical Particle | https://proofwiki.org/wiki/Kinetic_Energy_of_Classical_Particle | https://proofwiki.org/wiki/Kinetic_Energy_of_Classical_Particle | [
"Lagrangian Mechanics",
"Kinetic Energy"
] | [
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space/Real",
"Definition:Riemannian Manifold",
"Definition:Particle",
"Definition:Inertial Mass",
"Definition:Time",
"Definition:Variable/Real",
"Definition:Position",
"Definition:Differentiable Mapping",
"Definition:Dimension (Geometry)",
... | [] |
proofwiki-16433 | Principle of Stationary Action with Standard Lagrangian implies Newton's Laws of Motion | Let $\MM$ be an $n$-dimensional Euclidean manifold.
Let $P$ be a physical system composed of a countable number of classical particles with inertial masses $m_i$ with $i \in \N$.
Let $\mathbf x = \map {\mathbf x} t$ be twice-differentiable vector-valued function embedded in $\MM$.
Suppose ${\mathbf x}_i$ represents the... | Standard Lagrangian is of the following form:
:$\ds L = \sum_{i \mathop = 1}^n \frac {m_i} 2 \dot {\mathbf x}_i^2 - U$
By the Principle of Stationary Action, equations of motion of $P$ are:
:$\forall i \in \N: m_i {\ddot {\mathbf x} }_i + \dfrac {\partial U} {\partial \mathbf x_i} = 0$
By definition of velocity:
:${\do... | Let $\MM$ be an $n$-[[Definition:Dimension (Geometry)|dimensional]] [[Definition:Euclidean Space|Euclidean]] [[Definition:Riemannian Manifold|manifold]].
Let $P$ be a [[Definition:Physical System|physical system]] composed of a [[Definition:Countable Set|countable number]] of [[Definition:Classical Particle|classical ... | [[Definition:Standard Lagrangian|Standard Lagrangian]] is of the following form:
:$\ds L = \sum_{i \mathop = 1}^n \frac {m_i} 2 \dot {\mathbf x}_i^2 - U$
By the [[Principle of Stationary Action]], equations of motion of $P$ are:
:$\forall i \in \N: m_i {\ddot {\mathbf x} }_i + \dfrac {\partial U} {\partial \mathbf x... | Principle of Stationary Action with Standard Lagrangian implies Newton's Laws of Motion | https://proofwiki.org/wiki/Principle_of_Stationary_Action_with_Standard_Lagrangian_implies_Newton's_Laws_of_Motion | https://proofwiki.org/wiki/Principle_of_Stationary_Action_with_Standard_Lagrangian_implies_Newton's_Laws_of_Motion | [] | [
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space",
"Definition:Riemannian Manifold",
"Definition:Physical System",
"Definition:Countable Set",
"Definition:Classical Particle",
"Definition:Inertial Mass",
"Definition:Differentiability Class",
"Definition:Vector-Valued Function",
"Defi... | [
"Definition:Standard Lagrangian",
"Principle of Stationary Action",
"Definition:Velocity",
"Newton's Laws of Motion/Second Law"
] |
proofwiki-16434 | Altitudes of Triangle Bisect Angles of Orthic Triangle | Let $\triangle ABC$ be a triangle.
Let $\triangle DEF$ be its orthic triangle.
The altitudes of $\triangle ABC$ are the angle bisectors of $\triangle DEF$. | :400px
Consider the triangles $\triangle ABE$ and $\triangle ACF$.
We have that:
:$\angle FAC$ and $\angle BAE$ are common
:$\angle AFC$ and $\angle AEB$ are both right angles
and it follows from Triangles with Two Equal Angles are Similar that $\triangle ABE$ and $\triangle ACF$ are similar.
Thus:
:$\angle ABE = \angl... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle DEF$ be its [[Definition:Orthic Triangle|orthic triangle]].
The [[Definition:Altitude of Triangle|altitudes]] of $\triangle ABC$ are the [[Definition:Angle Bisector|angle bisectors]] of $\triangle DEF$. | :[[File:Pedal-Triangle-Angle-Bisectors.png|400px]]
Consider the [[Definition:Triangle (Geometry)|triangles]] $\triangle ABE$ and $\triangle ACF$.
We have that:
:$\angle FAC$ and $\angle BAE$ are common
:$\angle AFC$ and $\angle AEB$ are both [[Definition:Right Angle|right angles]]
and it follows from [[Triangles wit... | Altitudes of Triangle Bisect Angles of Orthic Triangle | https://proofwiki.org/wiki/Altitudes_of_Triangle_Bisect_Angles_of_Orthic_Triangle | https://proofwiki.org/wiki/Altitudes_of_Triangle_Bisect_Angles_of_Orthic_Triangle | [
"Orthic Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Orthic Triangle",
"Definition:Altitude of Triangle",
"Definition:Angle Bisector"
] | [
"File:Pedal-Triangle-Angle-Bisectors.png",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Triangles with Two Equal Angles are Similar",
"Definition:Similar Triangles",
"Definition:Quadrilateral",
"Definition:Right Angle",
"Definition:Polygon/Opposite",
"Definition:Right Angle",
"Oppo... |
proofwiki-16435 | Open Ball in Normed Division Ring is Open Ball in Induced Metric | Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the open ball in the normed division ring $\struct {R, \norm... | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} {a; \norm {\,\cdot\,} }
| o = \leadstoandfrom
| r = \norm {x - a} < \epsilon
| c = {{Defof|Open Ball of Normed Division Ring|Open Ball}} in $\struct {R, \norm {\,\cdot\,} }$
}}
{{eqn | o = \leadstoandfrom
| r = \map d {x, a} < \epsilon
| c ... | Let $\struct{R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly... | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} {a; \norm {\,\cdot\,} }
| o = \leadstoandfrom
| r = \norm {x - a} < \epsilon
| c = {{Defof|Open Ball of Normed Division Ring|Open Ball}} in $\struct {R, \norm {\,\cdot\,} }$
}}
{{eqn | o = \leadstoandfrom
| r = \map d {x, a} < \epsilon
| c ... | Open Ball in Normed Division Ring is Open Ball in Induced Metric | https://proofwiki.org/wiki/Open_Ball_in_Normed_Division_Ring_is_Open_Ball_in_Induced_Metric | https://proofwiki.org/wiki/Open_Ball_in_Normed_Division_Ring_is_Open_Ball_in_Induced_Metric | [
"Open Balls",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Open Ball/Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Open Ball",
"Definition:Metric Space"
] | [
"Definition:Set Equality"
] |
proofwiki-16436 | Closed Ball in Normed Division Ring is Closed Ball in Induced Metric | Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the closed ball in the normed division ring $\struct {R,... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}
| o = \le
| r = \epsilon
| c = {{Defof|Closed Ball of Normed Division Ring}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {x... | Let $\struct{R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}
| o = \le
| r = \epsilon
| c = {{Defof|Closed Ball of Normed Division Ring}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {x... | Closed Ball in Normed Division Ring is Closed Ball in Induced Metric | https://proofwiki.org/wiki/Closed_Ball_in_Normed_Division_Ring_is_Closed_Ball_in_Induced_Metric | https://proofwiki.org/wiki/Closed_Ball_in_Normed_Division_Ring_is_Closed_Ball_in_Induced_Metric | [
"Definitions/Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Closed Ball",
"Definition:Metric Space"
] | [
"Definition:Set Equality"
] |
proofwiki-16437 | Sphere in Normed Division Ring is Sphere in Induced Metric | Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the sphere in the normed division ring $\struct {R, \norm {\... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {S_\epsilon} {a; \norm {\,\cdot\,} }
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}
| r = \epsilon
| c = {{Defof|Sphere in Normed Division Ring}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {x, a}
| r = \epsilon
... | Let $\struct{R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {S_\epsilon} {a; \norm {\,\cdot\,} }
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - a}
| r = \epsilon
| c = {{Defof|Sphere in Normed Division Ring}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {x, a}
| r = \epsilon
... | Sphere in Normed Division Ring is Sphere in Induced Metric | https://proofwiki.org/wiki/Sphere_in_Normed_Division_Ring_is_Sphere_in_Induced_Metric | https://proofwiki.org/wiki/Sphere_in_Normed_Division_Ring_is_Sphere_in_Induced_Metric | [
"Definitions/Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Sphere/Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Sphere",
"Definition:Metric Space"
] | [
"Definition:Set Equality"
] |
proofwiki-16438 | Sphere is Set Difference of Closed Ball with Open Ball | Let $M = \struct{A, d}$ be a metric space or pseudometric space.
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} {a; d}$ denote the closed $\epsilon$-ball of $a$ in $M$.
Let $\map {B_\epsilon} {a; d}$ denote the open $\epsilon$-ball of $a$ in $M$.
Let $\map {S_\e... | {{begin-eqn}}
{{eqn | l = \map {S_\epsilon } a
| r = \set {x : \map d {x, a} = \epsilon}
| c = {{Defof|Sphere}}
}}
{{eqn | r = \set {x : \map d {x, a} \le \epsilon} \setminus \set {x : \map d {x, a} < \epsilon }
| c =
}}
{{eqn | r = \set {x : \map d {x, a} \le \epsilon} \setminus \map {B_\epsilon} a
... | Let $M = \struct{A, d}$ be a [[Definition:Metric Space|metric space]] or [[Definition:Pseudometric Space|pseudometric space]].
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map {{B_\epsilon}^-} {a; d}$ denote the [[Definition:Closed ... | {{begin-eqn}}
{{eqn | l = \map {S_\epsilon } a
| r = \set {x : \map d {x, a} = \epsilon}
| c = {{Defof|Sphere}}
}}
{{eqn | r = \set {x : \map d {x, a} \le \epsilon} \setminus \set {x : \map d {x, a} < \epsilon }
| c =
}}
{{eqn | r = \set {x : \map d {x, a} \le \epsilon} \setminus \map {B_\epsilon} a
... | Sphere is Set Difference of Closed Ball with Open Ball | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball | [
"Metric Spaces",
"Closed Balls",
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/Metric Space",
"Definition:Open Ball",
"Definition:Sphere"
] | [
"Category:Metric Spaces",
"Category:Closed Balls",
"Category:Open Balls"
] |
proofwiki-16439 | Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring | Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-closed ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$.
Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$... | The result follows directly from:
:Closed Ball in Normed Division Ring is Closed Ball in Induced Metric
:Open Ball in Normed Division Ring is Open Ball in Induced Metric
:Sphere in Normed Division Ring is Sphere in Induced Metric
:Sphere is Set Difference of Closed Ball with Open Ball
{{qed}}
Category:Normed Division R... | Let $\struct{R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the [[Definition:Closed Ball of No... | The result follows directly from:
:[[Closed Ball in Normed Division Ring is Closed Ball in Induced Metric]]
:[[Open Ball in Normed Division Ring is Open Ball in Induced Metric]]
:[[Sphere in Normed Division Ring is Sphere in Induced Metric]]
:[[Sphere is Set Difference of Closed Ball with Open Ball]]
{{qed}}
[[Categor... | Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball/Normed_Division_Ring | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball/Normed_Division_Ring | [
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/Normed Division Ring",
"Definition:Open Ball/Normed Division Ring",
"Definition:Sphere/Normed Division Ring"
] | [
"Closed Ball in Normed Division Ring is Closed Ball in Induced Metric",
"Open Ball in Normed Division Ring is Open Ball in Induced Metric",
"Sphere in Normed Division Ring is Sphere in Induced Metric",
"Sphere is Set Difference of Closed Ball with Open Ball",
"Category:Normed Division Rings"
] |
proofwiki-16440 | Conservation of Energy | Let $P$ be a physical system.
Let it have the action $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the standard Lagrangian, and $t$ is time.
Suppose $L$ does not depend on time explicitly:
:$\dfrac {\partial L} {\partial t} = 0$
Then the total energy of $P$ is conserved. | By assumption, $S$ is invariant under the following family of transformations:
:$T = t + \epsilon$
:$\mathbf X = \mathbf x$
By Noether's Theorem:
:$\nabla_{\dot {\mathbf x} } L \cdot \boldsymbol \psi + \paren {L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L} \phi = C$
where $\phi = 1$, $\boldsymbol \psi = \math... | Let $P$ be a [[Definition:Physical System|physical system]].
Let it have the [[Definition:Action of Physical System|action]] $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the [[Definition:Standard Lagrangian|standard Lagrangian]], and $t$ is [[Definition:Time|time]].
Suppose $L$ does not depend on [[Definit... | By assumption, $S$ is invariant under the following [[Definition:Family|family]] of [[Definition:Mapping|transformations]]:
:$T = t + \epsilon$
:$\mathbf X = \mathbf x$
By [[Noether's Theorem (Calculus of Variations)|Noether's Theorem]]:
:$\nabla_{\dot {\mathbf x} } L \cdot \boldsymbol \psi + \paren {L - \dot {\mat... | Conservation of Energy | https://proofwiki.org/wiki/Conservation_of_Energy | https://proofwiki.org/wiki/Conservation_of_Energy | [
"Conservation Laws",
"Energy",
"Lagrangian Mechanics",
"Physics"
] | [
"Definition:Physical System",
"Definition:Action of Physical System",
"Definition:Standard Lagrangian",
"Definition:Time",
"Definition:Time",
"Definition:Total Energy of Particles",
"Definition:Constant (Physics)"
] | [
"Definition:Family",
"Definition:Mapping",
"Noether's Theorem (Calculus of Variations)",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Total Energy of Particles",
"Definition:Constant (Physics)"
] |
proofwiki-16441 | Principle of Conservation of Linear Momentum | Let $P$ be a physical system.
Let it have the action $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the standard Lagrangian, and $t$ is time.
Suppose $L$ does not depend on one of the coordinates explicitly:
:$\dfrac {\partial L} {\partial x_j} = 0$
Then the total momentum of $P$ along the axis $x_j$ is conserve... | By assumption, $S$ is invariant under the following family of transformations:
:$T = t$
:$X_j = x_j + \epsilon$
:$X_{i \mathop \ne j} = x_{i \mathop \ne j}$
By Noether's Theorem:
:$\nabla_{\mathbf x} L \cdot \boldsymbol \psi + \paren {L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L} \phi = C$
where $\phi = 0$, ... | Let $P$ be a [[Definition:Physical System|physical system]].
Let it have the [[Definition:Action of Physical System|action]] $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the [[Definition:Standard Lagrangian|standard Lagrangian]], and $t$ is [[Definition:Time|time]].
Suppose $L$ does not depend on one of th... | By assumption, $S$ is invariant under the following [[Definition:Family|family]] of [[Definition:Mapping|transformations]]:
:$T = t$
:$X_j = x_j + \epsilon$
:$X_{i \mathop \ne j} = x_{i \mathop \ne j}$
By [[Noether's Theorem (Calculus of Variations)|Noether's Theorem]]:
:$\nabla_{\mathbf x} L \cdot \boldsymbol \ps... | Principle of Conservation of Linear Momentum | https://proofwiki.org/wiki/Principle_of_Conservation_of_Linear_Momentum | https://proofwiki.org/wiki/Principle_of_Conservation_of_Linear_Momentum | [
"Conservation Laws",
"Momentum",
"Lagrangian Mechanics",
"Physics"
] | [
"Definition:Physical System",
"Definition:Action of Physical System",
"Definition:Standard Lagrangian",
"Definition:Time",
"Definition:Linear Momentum",
"Definition:Axis/Coordinate Axes",
"Definition:Constant (Physics)"
] | [
"Definition:Family",
"Definition:Mapping",
"Noether's Theorem (Calculus of Variations)",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Momentum",
"Definition:Constant (Physics)"
] |
proofwiki-16442 | Conservation of Angular Momentum (Lagrangian Mechanics) | Let $P$ be a physical system composed of a finite number of particles.
Let $P$ have the action $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where:
:$L$ is the standard Lagrangian
:$t$ is time.
Let $L$ be invariant {{WRT}} rotation around the $z$-axis.
Then the total angular momentum of $P$ along the $z$-axis is conserved. | By assumption, $S$ is invariant under the following family of transformations:
{{begin-eqn}}
{{eqn | l = T
| r = t
}}
{{eqn | l = X_i
| r = x_i \cos \epsilon + y_i \sin \epsilon
}}
{{eqn | l = Y_i
| r = -x_i \sin \epsilon + y_i \cos \epsilon
}}
{{eqn | l = Z_i
| r = z_i
}}
{{end-eqn}}
where $\ep... | Let $P$ be a [[Definition:Physical System|physical system]] composed of a [[Definition:Finite Set|finite number]] of [[Definition:Particle|particles]].
Let $P$ have the [[Definition:Action of Physical System|action]] $S$:
:$\ds S = \int_{t_0}^{t_1} L \rd t$
where:
:$L$ is the [[Definition:Standard Lagrangian|standar... | By assumption, $S$ is [[Definition:Invariant|invariant]] under the following [[Definition:Family|family]] of [[Definition:Mapping|transformations]]:
{{begin-eqn}}
{{eqn | l = T
| r = t
}}
{{eqn | l = X_i
| r = x_i \cos \epsilon + y_i \sin \epsilon
}}
{{eqn | l = Y_i
| r = -x_i \sin \epsilon + y_i \co... | Conservation of Angular Momentum (Lagrangian Mechanics) | https://proofwiki.org/wiki/Conservation_of_Angular_Momentum_(Lagrangian_Mechanics) | https://proofwiki.org/wiki/Conservation_of_Angular_Momentum_(Lagrangian_Mechanics) | [
"Lagrangian Mechanics"
] | [
"Definition:Physical System",
"Definition:Finite Set",
"Definition:Particle",
"Definition:Action of Physical System",
"Definition:Standard Lagrangian",
"Definition:Time",
"Definition:Invariant",
"Definition:Rotation (Geometry)/Space",
"Definition:Axis/Z-Axis",
"Definition:Angular Momentum",
"Def... | [
"Definition:Invariant",
"Definition:Family",
"Definition:Mapping",
"Noether's Theorem (Calculus of Variations)",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Vector Quantity/Component/Z Component",
"Definition:Angular Momentum",
"Definition:Axis/Z-Axis",
"Definition:Constan... |
proofwiki-16443 | Leibniz's Law for Sets | Let $S$ be an arbitrary set.
Then:
:$x = y \dashv \vdash x \in S \iff y \in S$
for all $S$ in the universe of discourse.
This is therefore the justification behind the notion of the definition of set equality. | A direct application of Leibniz's law. | Let $S$ be an arbitrary [[Definition:Set|set]].
Then:
:$x = y \dashv \vdash x \in S \iff y \in S$
for all $S$ in the [[Definition:Universe of Discourse|universe of discourse]].
This is therefore the justification behind the notion of the definition of [[Definition:Set Equality/Definition 2|set equality]]. | A direct application of [[Axiom:Leibniz's Law|Leibniz's law]]. | Leibniz's Law for Sets | https://proofwiki.org/wiki/Leibniz's_Law_for_Sets | https://proofwiki.org/wiki/Leibniz's_Law_for_Sets | [
"Set Theory"
] | [
"Definition:Set",
"Definition:Universe of Discourse",
"Definition:Set Equality/Definition 2"
] | [
"Axiom:Leibniz's Law"
] |
proofwiki-16444 | Area of Annulus | Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$.
The area of $A$ is given by:
:$\map \Area A = \pi \paren {R^2 - r^2}$ | :400px
The area of $A$ is seen to be:
:the area of the outer circle with the area of the inner circle removed.
From Area of Circle:
:the area of the outer circle is $\pi R^2$
:the area of the inner circle is $\pi r^2$
The result follows.
{{qed}} | Let $A$ be an [[Definition:Annulus|annulus]] whose [[Definition:Inner Radius of Annulus|inner radius]] is $r$ and whose [[Definition:Outer Radius of Annulus|outer radius]] is $R$.
The [[Definition:Area|area]] of $A$ is given by:
:$\map \Area A = \pi \paren {R^2 - r^2}$ | :[[File:Annulus.png|400px]]
The [[Definition:Area|area]] of $A$ is seen to be:
:the [[Definition:Area|area]] of the outer [[Definition:Circle|circle]] with the [[Definition:Area|area]] of the inner [[Definition:Circle|circle]] removed.
From [[Area of Circle]]:
:the [[Definition:Area|area]] of the outer [[Definition:C... | Area of Annulus | https://proofwiki.org/wiki/Area_of_Annulus | https://proofwiki.org/wiki/Area_of_Annulus | [
"Annuli"
] | [
"Definition:Annulus",
"Definition:Annulus (Geometry)/Inner Radius",
"Definition:Annulus (Geometry)/Outer Radius",
"Definition:Area"
] | [
"File:Annulus.png",
"Definition:Area",
"Definition:Area",
"Definition:Circle",
"Definition:Area",
"Definition:Circle",
"Area of Circle",
"Definition:Area",
"Definition:Circle",
"Definition:Area",
"Definition:Circle"
] |
proofwiki-16445 | Area of Annulus as Area of Rectangle | Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$.
The area of $A$ is given by:
:$\map \Area A = 2 \pi \paren {r + \dfrac w 2} \times w$
where $w$ denotes the width of $A$.
That is, it is the area of the rectangle contained by:
:the width of $A$
:the circle midway in radius between the inner... | :400px
{{begin-eqn}}
{{eqn | l = \map \Area A
| r = \pi \paren {R^2 - r^2}
| c = Area of Annulus
}}
{{eqn | r = \pi \paren {\paren {r + w}^2 - r^2}
| c = {{Defof|Width of Annulus}}
}}
{{eqn | r = \pi \paren {r^2 + 2 r w + w^2 - r^2}
| c = Square of Sum
}}
{{eqn | r = \pi \paren {2 r + w} w
... | Let $A$ be an [[Definition:Annulus (Geometry)|annulus]] whose [[Definition:Inner Radius of Annulus|inner radius]] is $r$ and whose [[Definition:Outer Radius of Annulus|outer radius]] is $R$.
The [[Definition:Area|area]] of $A$ is given by:
:$\map \Area A = 2 \pi \paren {r + \dfrac w 2} \times w$
where $w$ denotes t... | :[[File:Annulus-mid-circle.png|400px]]
{{begin-eqn}}
{{eqn | l = \map \Area A
| r = \pi \paren {R^2 - r^2}
| c = [[Area of Annulus]]
}}
{{eqn | r = \pi \paren {\paren {r + w}^2 - r^2}
| c = {{Defof|Width of Annulus}}
}}
{{eqn | r = \pi \paren {r^2 + 2 r w + w^2 - r^2}
| c = [[Square of Sum]]
}}... | Area of Annulus as Area of Rectangle | https://proofwiki.org/wiki/Area_of_Annulus_as_Area_of_Rectangle | https://proofwiki.org/wiki/Area_of_Annulus_as_Area_of_Rectangle | [
"Annuli"
] | [
"Definition:Annulus (Geometry)",
"Definition:Annulus (Geometry)/Inner Radius",
"Definition:Annulus (Geometry)/Outer Radius",
"Definition:Area",
"Definition:Annulus (Geometry)/Width",
"Definition:Area",
"Definition:Quadrilateral/Rectangle",
"Definition:Quadrilateral/Rectangle/Containment",
"Definitio... | [
"File:Annulus-mid-circle.png",
"Area of Annulus",
"Square of Sum",
"Area of Parallelogram/Rectangle"
] |
proofwiki-16446 | Hamiltonian of Standard Lagrangian is Total Energy | Let $P$ be a physical system of classical particles.
Let $L$ be a standard Lagrangian associated with $P$.
Then the Hamiltonian of $P$ is the total energy of $P$. | {{begin-eqn}}
{{eqn | l = H
| r = -L + \sum_{i \mathop = 1}^n \dot {x_i} L_{\dot {x_i} }
}}
{{eqn | r = - \paren{T - V} + \sum_{i \mathop = 1}^n \dot {x_i} \dfrac {\partial} {\partial {\dot x}_i }\paren {\map T {\dot x} - \map V {t, x} }
| c = {{Defof|Standard Lagrangian}}
}}
{{eqn | r = -\paren{T - V} + \... | Let $P$ be a [[Definition:Physical System|physical system]] of [[Definition:Classical Particle|classical particles]].
Let $L$ be a [[Definition:Standard Lagrangian|standard Lagrangian]] associated with $P$.
Then the [[Definition:Hamiltonian|Hamiltonian]] of $P$ is the [[Definition:Total Energy of Particles|total ene... | {{begin-eqn}}
{{eqn | l = H
| r = -L + \sum_{i \mathop = 1}^n \dot {x_i} L_{\dot {x_i} }
}}
{{eqn | r = - \paren{T - V} + \sum_{i \mathop = 1}^n \dot {x_i} \dfrac {\partial} {\partial {\dot x}_i }\paren {\map T {\dot x} - \map V {t, x} }
| c = {{Defof|Standard Lagrangian}}
}}
{{eqn | r = -\paren{T - V} + \... | Hamiltonian of Standard Lagrangian is Total Energy | https://proofwiki.org/wiki/Hamiltonian_of_Standard_Lagrangian_is_Total_Energy | https://proofwiki.org/wiki/Hamiltonian_of_Standard_Lagrangian_is_Total_Energy | [
"Hamiltonians",
"Lagrangian Mechanics",
"Mathematical Physics",
"Physics"
] | [
"Definition:Physical System",
"Definition:Classical Particle",
"Definition:Standard Lagrangian",
"Definition:Hamiltonian",
"Definition:Total Energy of Particles"
] | [
"Kinetic Energy of Classical Particle",
"Kinetic Energy of Classical Particle",
"Definition:Total Energy of Particles"
] |
proofwiki-16447 | Characterization of Closed Ball in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {{B_\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\epsilon$.
Let $x, y \in \Q_p$.
Let $n \in \Z$.
{{TFAE}}
::$(1... | From P-adic Numbers form Non-Archimedean Valued Field:
:$\norm {\,\cdot\,}_p$ is a non-Archimedean norm. | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {{B_\epsilon}^-} a$ deno... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:$\norm {\,\cdot\,}_p$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]]. | Characterization of Closed Ball in P-adic Numbers | https://proofwiki.org/wiki/Characterization_of_Closed_Ball_in_P-adic_Numbers | https://proofwiki.org/wiki/Characterization_of_Closed_Ball_in_P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers/Center",
"Definition:Closed Ball/P-adic Numbers/Radius"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)"
] |
proofwiki-16448 | Characterization of Open Ball in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open ball of center $a$ of radius $\epsilon$.
Let $n \in \Z$.
Let $x, y \in \Q_p$.
{{TFAE}}
:$(1): \qua... | From P-adic Numbers form Non-Archimedean Valued Field:
:$\norm {\,\cdot\,}_p$ is a non-Archimedean norm. | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote t... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:$\norm {\,\cdot\,}_p$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]]. | Characterization of Open Ball in P-adic Numbers | https://proofwiki.org/wiki/Characterization_of_Open_Ball_in_P-adic_Numbers | https://proofwiki.org/wiki/Characterization_of_Open_Ball_in_P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Center",
"Definition:Open Ball/P-adic Numbers/Radius"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)"
] |
proofwiki-16449 | Poisson Brackets of Harmonic Oscillator | Let $P$ be a classical harmonic oscillator.
Let the real-valued function $\map x t$ be the position of $P$, where $t$ is time.
Then $P$ has the following Poisson brackets:
{{begin-eqn}}
{{eqn | l = \sqbrk {x, p}
| r = 1
}}
{{eqn | l = \sqbrk {x, H}
| r = \dfrac p m
}}
{{eqn | l = \sqbrk {p, H}
| r = -... | The standard Lagrangian of $P$ is:
:$L = \dfrac 1 2 \paren {m {\dot x}^2 - k x^2}$
The canonical momentum is:
:$p = \dfrac {\partial L} {\partial \dot x} = m \dot x$
The Hamiltonian associated to $L$ in canonical coordinates reads:
:$H = \dfrac {p^2} {2 m} + \dfrac k 2 x^2$
Then:
{{begin-eqn}}
{{eqn | l = \sqbrk {x, p}... | Let $P$ be a [[Definition:Classical Particle|classical]] [[Definition:Harmonic Oscillator|harmonic oscillator]].
Let the [[Definition:Real-Valued Function|real-valued function]] $\map x t$ be the [[Definition:Position|position]] of $P$, where $t$ is [[Definition:Time|time]].
Then $P$ has the following [[Definition:P... | The [[Definition:Standard Lagrangian|standard Lagrangian]] of $P$ is:
:$L = \dfrac 1 2 \paren {m {\dot x}^2 - k x^2}$
The [[Definition:Canonical Momentum|canonical momentum]] is:
:$p = \dfrac {\partial L} {\partial \dot x} = m \dot x$
The [[Definition:Hamiltonian|Hamiltonian]] associated to $L$ in [[Definition:Cano... | Poisson Brackets of Harmonic Oscillator | https://proofwiki.org/wiki/Poisson_Brackets_of_Harmonic_Oscillator | https://proofwiki.org/wiki/Poisson_Brackets_of_Harmonic_Oscillator | [
"Lagrangian Mechanics",
"Applied Mathematics",
"Mathematical Physics"
] | [
"Definition:Classical Particle",
"Definition:Harmonic Oscillator",
"Definition:Real-Valued Function",
"Definition:Position",
"Definition:Time",
"Definition:Poisson Bracket"
] | [
"Definition:Standard Lagrangian",
"Definition:Canonical Momentum",
"Definition:Hamiltonian",
"Definition:Canonical Coordinates"
] |
proofwiki-16450 | Equivalence of Definitions of Convergent P-adic Sequence | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence {x_n} $ be a sequence in $\Q_p$.
{{TFAE|def = Convergent P-adic Sequence|view = convergent $p$-adic sequence}}
=== Definition 1 ===
{{:Definition:Convergent Sequence/P-adic Numbers/Definition 1|Definition 1}}
==... | From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic norm is the norm on a division ring.
By definition, the $p$-adic metric is the metric induced by the $p$-adic norm.
From Equivalence of Definitions of Convergence in Normed Division Rings, it follows that
Definition 2, Definition 3 and Definition 4 a... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\sequence {x_n} $ be a [[Definition:Sequence|sequence]] in $\Q_p$.
{{TFAE|def = Convergent P-adic Sequence|view = convergent $p$-adic sequence}... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:the [[Definition:P-adic Norm|$p$-adic norm]] is the [[Definition:Norm on Division Ring|norm]] on a [[Definition:Division Ring|division ring]].
By definition, the [[Definition:P-adic Metric on P-adic Numbers|$p$-adic metric]] is the [[Definition:Metric Induced... | Equivalence of Definitions of Convergent P-adic Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_P-adic_Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_P-adic_Sequence | [
"Sequences",
"Convergence",
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Sequence",
"Definition:Convergent Sequence/P-adic Numbers/Definition 1",
"Definition:Convergent Sequence/P-adic Numbers/Definition 2",
"Definition:Convergent Sequence/P-adic Numbers/Definition 3",
"Definition:Convergent ... | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:P-adic Norm",
"Definition:Norm/Division Ring",
"Definition:Division Ring",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:P-adic Norm/P-adic Numbers",
"Equivalence... |
proofwiki-16451 | Equivalence of Definitions of Convergence in Normed Division Rings | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
{{TFAE|def = Convergent Sequence in Normed Division Ring}}
=== Definition 1 ===
{{:Definition:Convergent Sequence/Normed Division Ring/Definition 1}}
=== Definition 2 ===
{{:Definition:Convergent Sequence/Norm... | === Definition 1 iff Definition 2 ===
By definition, the metric induced by the norm $\norm {\, \cdot \,}$ is the mapping $d: R \times R \to \R_{\ge 0}$ defined as:
:$\map d {x, y} = \norm {x - y}$
From Metric Induced by Norm on Normed Division Ring is Metric, $d$ is a metric.
By definition of a convergent sequence in a... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|sequence in $R$]].
{{TFAE|def = Convergent Sequence in Normed Division Ring}}
=== [[Definition:Convergent Sequence/Normed Division Rin... | === Definition 1 iff Definition 2 ===
By definition, the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\, \cdot \,}$ is the [[Definition:Mapping|mapping]] $d: R \times R \to \R_{\ge 0}$ defined as:
:$\map d {x, y} = \norm {x - y}$
From [[Metric Induced by Norm on Normed Di... | Equivalence of Definitions of Convergence in Normed Division Rings | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergence_in_Normed_Division_Rings | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergence_in_Normed_Division_Rings | [
"Sequences",
"Convergence",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring/Definition 1",
"Definition:Convergent Sequence/Normed Division Ring/Definition 2",
"Definition:Convergent Sequence/Normed Division Ring/Definition 3",
"Definition:Conv... | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Mapping",
"Metric Induced by Norm on Normed Division Ring is Metric",
"Definition:Metric Space/Metric",
"Definition:Convergent Sequence/Metric Space",
"Definition:Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition... |
proofwiki-16452 | Finite Complement Topology is not Metrizable | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is not a metrizable space. | We have:
:Metrizable Space is Hausdorff
:Finite Complement Space is not $T_2$ (Hausdorff)
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
Then $T$ is not a [[Definition:Metrizable Space|metrizable space]]. | We have:
:[[Metrizable Space is Hausdorff]]
:[[Finite Complement Space is not T2|Finite Complement Space is not $T_2$ (Hausdorff)]]
Hence the result.
{{qed}} | Finite Complement Topology is not Metrizable | https://proofwiki.org/wiki/Finite_Complement_Topology_is_not_Metrizable | https://proofwiki.org/wiki/Finite_Complement_Topology_is_not_Metrizable | [
"Finite Complement Topologies",
"Examples of Metrizable Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:Metrizable Space"
] | [
"Metrizable Space is Hausdorff",
"Finite Complement Space is not T2"
] |
proofwiki-16453 | Poisson Brackets of Classical Particle in Radial Potential on Plane | Let $P$ be a classical particle embedded in a 2-dimensional Euclidean manifold.
Let the real-valued functions $\map r t$, $\map \theta t$ denote the position of $P$ in polar coordinates, where $t$ is time.
Suppose, the potential energy of $P$ depends only on $r$.
Then $P$ has the following Poisson brackets:
{{begin-eqn... | The standard Lagrangian of $P$ in polar coordinates is:
:$L = \dfrac 1 2 m \paren { {\dot r}^2 + r^2 {\dot \theta}^2 } - \map U r$
The canonical momenta are:
:$p_r = \dfrac {\partial L} {\partial \dot r} = m \dot r$
:$p_\theta = \dfrac {\partial L} {\partial \dot \theta} = m r^2 \dot \theta$
The Hamiltonian associated ... | Let $P$ be a [[Definition:Classical Particle|classical particle]] embedded in a 2-[[Definition:Dimension (Geometry)|dimensional]] [[Definition:Euclidean Space|Euclidean]] [[Definition:Riemannian Manifold|manifold]].
Let the [[Definition:Real-Valued Function|real-valued functions]] $\map r t$, $\map \theta t$ denote th... | The [[Definition:Standard Lagrangian|standard Lagrangian]] of $P$ in [[Definition:Polar Coordinates|polar coordinates]] is:
:$L = \dfrac 1 2 m \paren { {\dot r}^2 + r^2 {\dot \theta}^2 } - \map U r$
The [[Definition:Canonical Variable|canonical momenta]] are:
:$p_r = \dfrac {\partial L} {\partial \dot r} = m \dot r$... | Poisson Brackets of Classical Particle in Radial Potential on Plane | https://proofwiki.org/wiki/Poisson_Brackets_of_Classical_Particle_in_Radial_Potential_on_Plane | https://proofwiki.org/wiki/Poisson_Brackets_of_Classical_Particle_in_Radial_Potential_on_Plane | [
"Lagrangian Mechanics",
"Applied Mathematics",
"Mathematical Physics"
] | [
"Definition:Classical Particle",
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space",
"Definition:Riemannian Manifold",
"Definition:Real-Valued Function",
"Definition:Position",
"Definition:Polar Coordinates",
"Definition:Time",
"Definition:Potential Energy",
"Definition:Poisson Bracke... | [
"Definition:Standard Lagrangian",
"Definition:Polar Coordinates",
"Definition:Canonical Variable",
"Definition:Hamiltonian",
"Definition:Canonical Variable"
] |
proofwiki-16454 | Multiplication by Power of 10 by Moving Decimal Point | Let $n \in \R$ be a real number.
Let $n$ be expressed in decimal notation.
Let $10^d$ denote a power of $10$ for some integer $d$
Then $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $d$ places to the right.
Thus, if $d$ is negative, and so $10^d = 10^{-e}$ for some $e \in \Z_{>0}$, $... | Let $n$ be expressed in decimal notation as:
:$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0 \cdotp a_{-1} a_{-2} \dotso a_{-s} a_{-s - 1} \dotso}$
That is:
:$n = \ds \sum_{k \mathop \in \Z} a_k 10^k$
Then:
{{begin-eqn}}
{{eqn | l = n \times 10^d
| r = 10^d \times \sum_{k \mathop \in \Z} a_k 10^k
| c =
}}
{{eqn... | Let $n \in \R$ be a [[Definition:Real Number|real number]].
Let $n$ be expressed in [[Definition:Decimal Notation|decimal notation]].
Let $10^d$ denote a [[Definition:Integer Power|power of $10$]] for some [[Definition:Integer|integer]] $d$
Then $n \times 10^d$ can be expressed in [[Definition:Decimal Notation|dec... | Let $n$ be expressed in [[Definition:Decimal Notation|decimal notation]] as:
:$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0 \cdotp a_{-1} a_{-2} \dotso a_{-s} a_{-s - 1} \dotso}$
That is:
:$n = \ds \sum_{k \mathop \in \Z} a_k 10^k$
Then:
{{begin-eqn}}
{{eqn | l = n \times 10^d
| r = 10^d \times \sum_{k \mathop \i... | Multiplication by Power of 10 by Moving Decimal Point | https://proofwiki.org/wiki/Multiplication_by_Power_of_10_by_Moving_Decimal_Point | https://proofwiki.org/wiki/Multiplication_by_Power_of_10_by_Moving_Decimal_Point | [
"Decimal Notation"
] | [
"Definition:Real Number",
"Definition:Decimal Notation",
"Definition:Power (Algebra)/Integer",
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Decimal Expansion/Decimal Point",
"Definition:Right (Direction)",
"Definition:Negative/Integer",
"Definition:Decimal Notation",
"Definitio... | [
"Definition:Decimal Notation",
"Distributive Laws/Arithmetic",
"Exponent Combination Laws/Product of Powers",
"Translation of Index Variable of Summation",
"Definition:Digit",
"Definition:Decimal Expansion/Decimal Point",
"Definition:Positive/Integer",
"Definition:Decimal Expansion/Decimal Point",
"... |
proofwiki-16455 | Number of Significant Figures in Result of Multiplication | Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively.
Then the most significant figures that $m \times n$ can have is $\min \set {d_m, d_n}$. | {{ProofWanted|I need a run-up.}} | Let $m$ and $n$ be [[Definition:Number|numbers]] which are presented to $d_m$ and $d_n$ [[Definition:Significant Figures|significant figures]] respectively.
Then the most [[Definition:Significant Figures|significant figures]] that $m \times n$ can have is $\min \set {d_m, d_n}$. | {{ProofWanted|I need a run-up.}} | Number of Significant Figures in Result of Multiplication | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Multiplication | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Multiplication | [
"Significant Figures"
] | [
"Definition:Number",
"Definition:Significant Figures",
"Definition:Significant Figures"
] | [] |
proofwiki-16456 | Number of Significant Figures in Result of Division | Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively.
Then the most significant figures that $\dfrac m n$ can have is $\min \set {d_m, d_n}$. | {{ProofWanted|I need a run-up.}} | Let $m$ and $n$ be [[Definition:Number|numbers]] which are presented to $d_m$ and $d_n$ [[Definition:Significant Figures|significant figures]] respectively.
Then the most [[Definition:Significant Figures|significant figures]] that $\dfrac m n$ can have is $\min \set {d_m, d_n}$. | {{ProofWanted|I need a run-up.}} | Number of Significant Figures in Result of Division | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Division | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Division | [
"Significant Figures"
] | [
"Definition:Number",
"Definition:Significant Figures",
"Definition:Significant Figures"
] | [] |
proofwiki-16457 | Number of Significant Figures in Result of Square Root | Let $m$ be a number which is presented to $d$ significant figures.
Then the most significant figures that $\sqrt m$ can have is also $d$. | {{ProofWanted|I need a run-up.}} | Let $m$ be a [[Definition:Number|number]] which is presented to $d$ [[Definition:Significant Figures|significant figures]].
Then the most [[Definition:Significant Figures|significant figures]] that $\sqrt m$ can have is also $d$. | {{ProofWanted|I need a run-up.}} | Number of Significant Figures in Result of Square Root | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Square_Root | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Square_Root | [
"Significant Figures"
] | [
"Definition:Number",
"Definition:Significant Figures",
"Definition:Significant Figures"
] | [] |
proofwiki-16458 | Number of Significant Figures in Result of Addition or Subtraction | Let $m$ and $n$ be numbers.
Let $d_m$ and $d_n$ be the position of the {{LSD}} of $m$ and $n$ respectively.
Then the {{LSD}} in either $m + n$ or $m - n$ is in the position corresponding to the greater significant digit of $d_m$ and $d_n$. | {{ProofWanted|I need a run-up.}} | Let $m$ and $n$ be [[Definition:Number|numbers]].
Let $d_m$ and $d_n$ be the position of the {{LSD}} of $m$ and $n$ respectively.
Then the {{LSD}} in either $m + n$ or $m - n$ is in the position corresponding to the greater [[Definition:Significant Figures|significant digit]] of $d_m$ and $d_n$. | {{ProofWanted|I need a run-up.}} | Number of Significant Figures in Result of Addition or Subtraction | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Addition_or_Subtraction | https://proofwiki.org/wiki/Number_of_Significant_Figures_in_Result_of_Addition_or_Subtraction | [
"Significant Figures"
] | [
"Definition:Number",
"Definition:Significant Figures"
] | [] |
proofwiki-16459 | Range of Common Logarithm of Number between 1 and 10 | Let $x \in \R$ be a real number such that:
:$1 \le x < 10$
Then:
:$0 \le \log_{10} x \le 1$
where $\log_{10}$ denotes the common logarithm function. | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = 10^0
| c = {{Defof|Integer Power}}
}}
{{eqn | l = 10
| r = 10^1
| c = {{Defof|Integer Power}}
}}
{{eqn | ll= \leadsto
| l = \log_{10} 1
| r = 0
| c =
}}
{{eqn | l = \log_{10} 10
| r = 1
| c =
}}
{{end-eqn}}
The result fol... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that:
:$1 \le x < 10$
Then:
:$0 \le \log_{10} x \le 1$
where $\log_{10}$ denotes the [[Definition:Common Logarithm|common logarithm]] function. | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = 10^0
| c = {{Defof|Integer Power}}
}}
{{eqn | l = 10
| r = 10^1
| c = {{Defof|Integer Power}}
}}
{{eqn | ll= \leadsto
| l = \log_{10} 1
| r = 0
| c =
}}
{{eqn | l = \log_{10} 10
| r = 1
| c =
}}
{{end-eqn}}
The result f... | Range of Common Logarithm of Number between 1 and 10 | https://proofwiki.org/wiki/Range_of_Common_Logarithm_of_Number_between_1_and_10 | https://proofwiki.org/wiki/Range_of_Common_Logarithm_of_Number_between_1_and_10 | [
"Common Logarithms"
] | [
"Definition:Real Number",
"Definition:General Logarithm/Common"
] | [
"Logarithm is Strictly Increasing"
] |
proofwiki-16460 | Common Logarithm of Number in Scientific Notation | Let $n$ be a positive real number which is presented (possibly approximated) in scientific notation as:
:$n = a \times 10^d$
where:
:$1 \le a < 10$
:$d \in \Z$ is an integer.
Then:
:$\log_{10} n = \log_{10} a + d$
where:
:$0 \le \log_{10} a < 1$ | {{begin-eqn}}
{{eqn | l = n
| r = a \times 10^d
| c = by definition
}}
{{eqn | ll= \leadsto
| l = \log_{10} n
| r = \map {\log_{10} } {a \times 10^d}
| c =
}}
{{eqn | r = \log_{10} a + \log_{10} 10^d
| c = Logarithm of Product
}}
{{eqn | r = \log_{10} a + d
| c = {{Defof|Commo... | Let $n$ be a [[Definition:Positive Real Number|positive real number]] which is presented (possibly approximated) in [[Definition:Scientific Notation|scientific notation]] as:
:$n = a \times 10^d$
where:
:$1 \le a < 10$
:$d \in \Z$ is an [[Definition:Integer|integer]].
Then:
:$\log_{10} n = \log_{10} a + d$
where:
:... | {{begin-eqn}}
{{eqn | l = n
| r = a \times 10^d
| c = by definition
}}
{{eqn | ll= \leadsto
| l = \log_{10} n
| r = \map {\log_{10} } {a \times 10^d}
| c =
}}
{{eqn | r = \log_{10} a + \log_{10} 10^d
| c = [[Logarithm of Product]]
}}
{{eqn | r = \log_{10} a + d
| c = {{Defof|C... | Common Logarithm of Number in Scientific Notation | https://proofwiki.org/wiki/Common_Logarithm_of_Number_in_Scientific_Notation | https://proofwiki.org/wiki/Common_Logarithm_of_Number_in_Scientific_Notation | [
"Common Logarithms"
] | [
"Definition:Positive/Real Number",
"Definition:Scientific Notation",
"Definition:Integer"
] | [
"Sum of Logarithms",
"Range of Common Logarithm of Number between 1 and 10",
"Category:Common Logarithms"
] |
proofwiki-16461 | Characteristic of Common Logarithm of Number Greater than 1 | Let $x \in \R_{>1}$ be a (strictly) positive real number greater than $1$.
The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of digits to the left of the decimal point of $x$. | Let $x$ be expressed in scientific notation:
:$x = a \times 10^e$
where:
:$1 \le a < 10$
:$e \in \Z_{\ge 0}$
From Range of Common Logarithm of Number between 1 and 10:
:$0 \le \log_{10} a < 1$
The characteristic of $\log_{10} x$ equals $\map {\log_{10} } {10^e} = e$.
Thus the characteristic of $\log_{10} x$ is equal to... | Let $x \in \R_{>1}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] greater than $1$.
The [[Definition:Characteristic of Common Logarithm|characteristic]] of its [[Definition:Common Logarithm|common logarithm]] $\log_{10} x$ is equal to one less than the number of [[Definition:Digit|d... | Let $x$ be expressed in [[Definition:Scientific Notation|scientific notation]]:
:$x = a \times 10^e$
where:
:$1 \le a < 10$
:$e \in \Z_{\ge 0}$
From [[Range of Common Logarithm of Number between 1 and 10]]:
:$0 \le \log_{10} a < 1$
The [[Definition:Characteristic of Common Logarithm|characteristic]] of $\log_{10} x$... | Characteristic of Common Logarithm of Number Greater than 1 | https://proofwiki.org/wiki/Characteristic_of_Common_Logarithm_of_Number_Greater_than_1 | https://proofwiki.org/wiki/Characteristic_of_Common_Logarithm_of_Number_Greater_than_1 | [
"Common Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:General Logarithm/Common/Characteristic",
"Definition:General Logarithm/Common",
"Definition:Digit",
"Definition:Left (Direction)",
"Definition:Decimal Expansion/Decimal Point"
] | [
"Definition:Scientific Notation",
"Range of Common Logarithm of Number between 1 and 10",
"Definition:General Logarithm/Common/Characteristic",
"Definition:General Logarithm/Common/Characteristic",
"Definition:Scientific Notation/Exponent",
"Multiplication by Power of 10 by Moving Decimal Point",
"Defin... |
proofwiki-16462 | Characteristic of Common Logarithm of Number Less than 1 | Let $x \in \R_{>0}$ be a (strictly) positive real number such that $x < 1$.
The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of zero digits to the immediate right of the decimal point of $x$. | Let $x$ be expressed in scientific notation:
:$x = a \times 10^{-e}$
where:
:$1 \le a < 10$
:$e \in \Z_{>0}$
From Range of Common Logarithm of Number between 1 and 10:
:$0 \le \log_{10} a < 1$
The characteristic of $\log_{10} x$ equals $\map {\log_{10} } {10^{-e} } = \overline e$.
Thus the characteristic of $\log_{10} ... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] such that $x < 1$.
The [[Definition:Characteristic of Common Logarithm|characteristic]] of its [[Definition:Common Logarithm|common logarithm]] $\log_{10} x$ is equal to one less than the number of [[Definition:Zero D... | Let $x$ be expressed in [[Definition:Scientific Notation|scientific notation]]:
:$x = a \times 10^{-e}$
where:
:$1 \le a < 10$
:$e \in \Z_{>0}$
From [[Range of Common Logarithm of Number between 1 and 10]]:
:$0 \le \log_{10} a < 1$
The [[Definition:Characteristic of Common Logarithm|characteristic]] of $\log_{10} x$... | Characteristic of Common Logarithm of Number Less than 1 | https://proofwiki.org/wiki/Characteristic_of_Common_Logarithm_of_Number_Less_than_1 | https://proofwiki.org/wiki/Characteristic_of_Common_Logarithm_of_Number_Less_than_1 | [
"Common Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:General Logarithm/Common/Characteristic",
"Definition:General Logarithm/Common",
"Definition:Zero Digit",
"Definition:Right (Direction)",
"Definition:Decimal Expansion/Decimal Point"
] | [
"Definition:Scientific Notation",
"Range of Common Logarithm of Number between 1 and 10",
"Definition:General Logarithm/Common/Characteristic",
"Definition:General Logarithm/Common/Characteristic",
"Definition:Scientific Notation/Exponent",
"Definition:Digit",
"Definition:Decimal Expansion/Decimal Point... |
proofwiki-16463 | Intersection of Plane with Sphere is Circle | The intersection of a plane with a sphere is a circle. | Let $S$ be a sphere of radius $R$ whose center is located for convenience at the origin.
Let $P$ be a plane which intersects $S$ but is not a tangent plane to $S$.
It is to be shown that $S \cap P$ is a circle.
Let $S$ and $P$ be embedded in a (real) cartesian space of $3$ dimensions.
Let this space be rotated until $P... | The [[Definition:Intersection (Geometry)|intersection]] of a [[Definition:Plane|plane]] with a [[Definition:Sphere (Geometry)|sphere]] is a [[Definition:Circle|circle]]. | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $R$ whose [[Definition:Center of Sphere|center]] is located for convenience at the [[Definition:Origin|origin]].
Let $P$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $S$ but is not ... | Intersection of Plane with Sphere is Circle | https://proofwiki.org/wiki/Intersection_of_Plane_with_Sphere_is_Circle | https://proofwiki.org/wiki/Intersection_of_Plane_with_Sphere_is_Circle | [
"Spheres"
] | [
"Definition:Intersection (Geometry)",
"Definition:Plane Surface",
"Definition:Sphere/Geometry",
"Definition:Circle"
] | [
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Sphere/Geometry/Center",
"Definition:Coordinate System/Origin",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Tangent Plane",
"Definition:Circle",
"Definition:Cartesian Product/Cartesian Sp... |
proofwiki-16464 | P-adic Norm satisfies Non-Archimedean Norm Axioms | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \to \R_{\ge 0}$.
Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms:
{{begin-axiom}}
{{axiom | n = \text N 1
| lc= Positive Definiteness:
| q = \f... | From P-adic Numbers form Non-Archimedean Valued Field:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a non-Archimedean norm
{{qed}}
Category:P-adic Number Theory
9913avh0grele7l08r7j9381lrlbw9z | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] with [[Definition:P-adic Norm on P-adic Numbers|$p$-adic norm]] $\norm {\,\cdot\,}_p : \Q_p \to \R_{\ge 0}$.
Then $\norm {\,\cdot\,}_p$ satisfies the ... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]]
{{qed}}
[[Category:P-adic Number Theory]]
9913avh0grele7l08r7j9381lrlbw9z | P-adic Norm satisfies Non-Archimedean Norm Axioms | https://proofwiki.org/wiki/P-adic_Norm_satisfies_Non-Archimedean_Norm_Axioms | https://proofwiki.org/wiki/P-adic_Norm_satisfies_Non-Archimedean_Norm_Axioms | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Norm/P-adic Numbers",
"Axiom:Non-Archimedean Norm Axioms/Division Ring",
"Definition:Positive Definite (Ring)",
"Definition:Multiplicative Function on Ring"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Category:P-adic Number Theory"
] |
proofwiki-16465 | Circles with Same Poles are Parallel | Let $S$ be a sphere.
Let $C$ and $D$ be circles on $S$ (either great circles or small circles).
Let $C$ and $D$ both have the same pair of poles.
Then $C$ and $D$ are parallel. | Let the poles of $C$ and $D$ be $A$ and $B$.
$C$ and $D$ each lie embedded in a plane.
Both of these planes by definition are perpendicular to $AB$.
The result follows from Planes Perpendicular to same Straight Line are Parallel.
{{qed}} | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]].
Let $C$ and $D$ be [[Definition:Circle|circles]] on $S$ (either [[Definition:Great Circle|great circles]] or [[Definition:Small Circle|small circles]]).
Let $C$ and $D$ both have the same pair of [[Definition:Pole of Circle|poles]].
Then $C$ and $D$ are [[Definit... | Let the [[Definition:Pole of Circle|poles]] of $C$ and $D$ be $A$ and $B$.
$C$ and $D$ each lie embedded in a [[Definition:Plane|plane]].
Both of these [[Definition:Plane|planes]] by definition are [[Definition:Line Perpendicular to Plane|perpendicular]] to $AB$.
The result follows from [[Planes Perpendicular to sam... | Circles with Same Poles are Parallel | https://proofwiki.org/wiki/Circles_with_Same_Poles_are_Parallel | https://proofwiki.org/wiki/Circles_with_Same_Poles_are_Parallel | [
"Spherical Geometry"
] | [
"Definition:Sphere/Geometry",
"Definition:Circle",
"Definition:Great Circle",
"Definition:Small Circle",
"Definition:Pole of Circle",
"Definition:Parallel (Geometry)/Planes"
] | [
"Definition:Pole of Circle",
"Definition:Plane Surface",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Planes Perpendicular to same Straight Line are Parallel"
] |
proofwiki-16466 | Three Points on Sphere in Same Hemisphere | Let $S$ be a sphere.
Let $A$, $B$ and $C$ be points on $S$ which do not all lie on the same great circle.
Then it is possible to divide $S$ into two hemispheres such that $A$, $B$ and $C$ all lie on the same hemisphere. | Because $A$, $B$ and $C$ do not lie on the same great circle, no two of these points are the endpoints of the same diameter of $S$.
Otherwise it would be possible to construct a great circle passing through all $3$ points $A$, $B$ and $C$.
Let a great circle $E$ be constructed through $A$ and $B$.
Then as $C$ is not on... | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]].
Let $A$, $B$ and $C$ be [[Definition:Point|points]] on $S$ which do not all lie on the same [[Definition:Great Circle|great circle]].
Then it is possible to divide $S$ into two [[Definition:Hemisphere|hemispheres]] such that $A$, $B$ and $C$ all lie on the same [... | Because $A$, $B$ and $C$ do not lie on the same [[Definition:Great Circle|great circle]], no two of these [[Definition:Point|points]] are the [[Definition:Endpoint of Line|endpoints]] of the same [[Definition:Diameter of Sphere|diameter]] of $S$.
Otherwise it would be possible to construct a [[Definition:Great Circle|... | Three Points on Sphere in Same Hemisphere | https://proofwiki.org/wiki/Three_Points_on_Sphere_in_Same_Hemisphere | https://proofwiki.org/wiki/Three_Points_on_Sphere_in_Same_Hemisphere | [
"Spherical Geometry"
] | [
"Definition:Sphere/Geometry",
"Definition:Point",
"Definition:Great Circle",
"Definition:Hemisphere",
"Definition:Hemisphere"
] | [
"Definition:Great Circle",
"Definition:Point",
"Definition:Line/Endpoint",
"Definition:Sphere/Geometry/Diameter",
"Definition:Great Circle",
"Definition:Point",
"Definition:Great Circle",
"Definition:Spherical Angle",
"Definition:Sphere/Geometry/Diameter",
"Definition:Line/Endpoint",
"Definition... |
proofwiki-16467 | Side of Spherical Triangle is Less than 2 Right Angles | Let $ABC$ be a spherical triangle on a sphere $S$.
Let $AB$ be a side of $ABC$.
The '''length''' of $AB$ is less than $2$ right angles. | $A$ and $B$ are two points on a great circle $E$ of $S$ which are not both on the same diameter.
So $AB$ is not equal to $2$ right angles.
Then it is noted that both $A$ and $B$ are in the same hemisphere, from Three Points on Sphere in Same Hemisphere.
That means the distance along $E$ is less than one semicircle of $... | Let $ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on a [[Definition:Sphere (Geometry)|sphere]] $S$.
Let $AB$ be a [[Definition:Side of Spherical Triangle|side]] of $ABC$.
The '''[[Definition:Length of Side of Spherical Triangle|length]]''' of $AB$ is less than $2$ [[Definition:Right Angle|right ang... | $A$ and $B$ are two [[Definition:Point|points]] on a [[Definition:Great Circle|great circle]] $E$ of $S$ which are not both on the same [[Definition:Diameter of Sphere|diameter]].
So $AB$ is not equal to $2$ [[Definition:Right Angle|right angles]].
Then it is noted that both $A$ and $B$ are in the same [[Definition:H... | Side of Spherical Triangle is Less than 2 Right Angles | https://proofwiki.org/wiki/Side_of_Spherical_Triangle_is_Less_than_2_Right_Angles | https://proofwiki.org/wiki/Side_of_Spherical_Triangle_is_Less_than_2_Right_Angles | [
"Spherical Triangles"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle/Side",
"Definition:Spherical Triangle/Side/Length",
"Definition:Right Angle"
] | [
"Definition:Point",
"Definition:Great Circle",
"Definition:Sphere/Geometry/Diameter",
"Definition:Right Angle",
"Definition:Hemisphere",
"Three Points on Sphere in Same Hemisphere",
"Definition:Circle/Semicircle",
"Definition:Spherical Angle",
"Definition:Spherical Triangle/Side/Length"
] |
proofwiki-16468 | Center is Element of Closed Ball | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a positive real number.
Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $M$.
Then:
:$a \in \map {{B_\epsilon}^-} a$ | By metric axiom $(\text M 1)$:
:$\map d {a, a} = 0$
By assumption:
:$\epsilon > 0$
Hence:
:$\map d {a, a} < \epsilon$
By definition of the closed $\epsilon$-ball of $a$ $\map {{B_\epsilon}^-} a$ in $M$:
:$a \in \map { {B_\epsilon}^-} a$
{{qed}}
Category:Metric Spaces
8qmx694flnod8zy7dm2tndr6ifttdir | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Positive Real Number|positive real number]].
Let $\map { {B_\epsilon}^-} a$ be the [[Definition:Closed Ball|closed $\epsilon$-ball of $a$]] in $M$.
Then:
:$a \in \map {{B_\epsilon}^-}... | By [[Definition:Metric Space|metric axiom $(\text M 1)$]]:
:$\map d {a, a} = 0$
By assumption:
:$\epsilon > 0$
Hence:
:$\map d {a, a} < \epsilon$
By definition of the [[Definition:Closed Ball|closed $\epsilon$-ball of $a$]] $\map {{B_\epsilon}^-} a$ in $M$:
:$a \in \map { {B_\epsilon}^-} a$
{{qed}}
[[Category:Metri... | Center is Element of Closed Ball | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Positive/Real Number",
"Definition:Closed Ball"
] | [
"Definition:Metric Space",
"Definition:Closed Ball",
"Category:Metric Spaces"
] |
proofwiki-16469 | Center is Element of Closed Ball/Normed Division Ring | Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$.
Then:
:$a \in \map { {B_\epsilon}^-} a$ | Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
From Closed Ball in Normed Division Ring is Closed Ball in Induced Metric, $\map { {B_\epsilon}^-} a$ is the closed $\epsilon$-ball of $a$ in the metric space $\struct{R,d}$.
From Center is Element of Closed Ball:
:$a \in \map { {B_\epsilon}^-} a$
{{qed}}
C... | Let $\struct{R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map { {B_\epsilon}^-} a$ be the [[Definition:Closed Ball of Normed Division Ring|closed ... | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
From [[Closed Ball in Normed Division Ring is Closed Ball in Induced Metric]], $\map { {B_\epsilon}^-} a$ is the [[Definition:Closed Ball|closed $\epsilon$-ball of ... | Center is Element of Closed Ball/Normed Division Ring | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball/Normed_Division_Ring | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball/Normed_Division_Ring | [
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Closed Ball in Normed Division Ring is Closed Ball in Induced Metric",
"Definition:Closed Ball",
"Definition:Metric Space",
"Center is Element of Closed Ball",
"Category:Normed Division Rings"
] |
proofwiki-16470 | Center is Element of Closed Ball/P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
Then:
:$a \in \map { {B_\epsilon}^-... | By definition, $\map { {B_\epsilon}^-} a$ is the closed $\epsilon$-ball of $a$ in the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
From Center is Element of Closed Ball in Normed Division Ring:
$a \in \map { {B_\epsilon}^-} a$
{{qed}}
Category:Topology of P-adic Numbers
rvja6236q3kwg9jopoul6hj4l6rd5yw | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map { {B_\epsilo... | By definition, $\map { {B_\epsilon}^-} a$ is the [[Definition:Closed Ball of Normed Division Ring|closed $\epsilon$-ball of $a$]] in the [[Definition:Normed Division Ring|normed division ring]] $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
From [[Center is Element of Closed Ball in Normed Division Ring]]:
$a \in \map { {B_\e... | Center is Element of Closed Ball/P-adic Numbers | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball/P-adic_Numbers | https://proofwiki.org/wiki/Center_is_Element_of_Closed_Ball/P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/P-adic Numbers"
] | [
"Definition:Closed Ball/Normed Division Ring",
"Definition:Normed Division Ring",
"Center is Element of Closed Ball/Normed Division Ring",
"Category:Topology of P-adic Numbers"
] |
proofwiki-16471 | Length of Arc of Small Circle | Let $S$ be a sphere.
Let $\bigcirc FCD$ be a small circle on $S$.
Let $C$ and $D$ be the points on $\bigcirc FCD$ such that $CD$ is the arc of $\bigcirc FCD$ whose length is to be determined. | :500px
Let $R$ denote the center of $\bigcirc FCD$.
Let $O$ denote the center of $S$, which is also the center of $\bigcirc EAB$.
We have:
:$CD = RC \times \angle CRD$
Similarly:
:$AB = OA \times \angle AOB$
By Circles with Same Poles are Parallel:
:$\bigcirc FCD \parallel \bigcirc EAB$
Hence $RC$ and $RD$ are parallel... | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]].
Let $\bigcirc FCD$ be a [[Definition:Small Circle|small circle]] on $S$.
Let $C$ and $D$ be the [[Definition:Point|points]] on $\bigcirc FCD$ such that $CD$ is the [[Definition:Arc of Circle|arc]] of $\bigcirc FCD$ whose [[Definition:Arc Length|length]] is to be d... | :[[File:Size-of-small-circle.png|500px]]
Let $R$ denote the [[Definition:Center of Circle|center]] of $\bigcirc FCD$.
Let $O$ denote the [[Definition:Center of Sphere|center]] of $S$, which is also the [[Definition:Center of Circle|center]] of $\bigcirc EAB$.
We have:
:$CD = RC \times \angle CRD$
Similarly:
:$AB = ... | Length of Arc of Small Circle | https://proofwiki.org/wiki/Length_of_Arc_of_Small_Circle | https://proofwiki.org/wiki/Length_of_Arc_of_Small_Circle | [
"Small Circles"
] | [
"Definition:Sphere/Geometry",
"Definition:Small Circle",
"Definition:Point",
"Definition:Circle/Arc",
"Definition:Arc Length",
"Definition:Point",
"Definition:Arc Length",
"Definition:Circle/Arc"
] | [
"File:Size-of-small-circle.png",
"Definition:Circle/Center",
"Definition:Sphere/Geometry/Center",
"Definition:Circle/Center",
"Circles with Same Poles are Parallel",
"Definition:Parallel (Geometry)/Lines",
"Definition:Sphere/Geometry/Radius",
"Definition:Angle",
"Definition:Subtend",
"Definition:C... |
proofwiki-16472 | Distance Between Points of Same Latitude along Parallel of Latitude | Let $J$ and $K$ be points on Earth's surface that have the same latitude.
Let $JK$ be the length of the arc joining $JK$ measured along the parallel of latitude on which they both lie.
Let $R$ denote the center of the parallel of latitude holding $J$ and $K$.
Let $\operatorname {Long}_J$ and $\operatorname {Long}_K$ de... | We have that $\size {\operatorname {Long}_J - \operatorname {Long}_K}$ is the magnitude of the spherical angle between the meridians on which $J$ and $K$ lie.
Thus from Length of Arc of Small Circle:
:$JK \approx \size {\operatorname {Long}_J - \operatorname {Long}_K} \cos \operatorname {Lat}_J$ degrees of arc
Then by ... | Let $J$ and $K$ be [[Definition:Point|points]] on [[Definition:Earth|Earth's]] surface that have the same [[Definition:Terrestrial Latitude|latitude]].
Let $JK$ be the [[Definition:Arc Length|length]] of the [[Definition:Arc of Circle|arc]] joining $JK$ measured along the [[Definition:Parallel of Latitude|parallel of ... | We have that $\size {\operatorname {Long}_J - \operatorname {Long}_K}$ is the [[Definition:Magnitude|magnitude]] of the [[Definition:Spherical Angle|spherical angle]] between the [[Definition:Terrestrial Meridian|meridians]] on which $J$ and $K$ lie.
Thus from [[Length of Arc of Small Circle]]:
:$JK \approx \size {\op... | Distance Between Points of Same Latitude along Parallel of Latitude | https://proofwiki.org/wiki/Distance_Between_Points_of_Same_Latitude_along_Parallel_of_Latitude | https://proofwiki.org/wiki/Distance_Between_Points_of_Same_Latitude_along_Parallel_of_Latitude | [
"Geodesy"
] | [
"Definition:Point",
"Definition:Earth",
"Definition:Latitude/Terrestrial",
"Definition:Arc Length",
"Definition:Circle/Arc",
"Definition:Parallel of Latitude",
"Definition:Circle/Center",
"Definition:Parallel of Latitude",
"Definition:Longitude/Terrestrial",
"Definition:Degree of Arc",
"Definiti... | [
"Definition:Magnitude",
"Definition:Spherical Angle",
"Definition:Meridian/Terrestrial",
"Length of Arc of Small Circle",
"Definition:Degree of Arc",
"Definition:Nautical Mile",
"Definition:Nautical Mile",
"Definition:Degree of Arc",
"Definition:Latitude/Terrestrial",
"Definition:Meridian/Terrestr... |
proofwiki-16473 | Center is Element of Open Ball | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a positive real number.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $M$.
Then:
:$a \in \map {B_\epsilon} a$ | By {{Metric-space-axiom|1}}:
:$\map d {a, a} = 0$
By assumption:
:$\epsilon > 0$
Hence:
:$\map d {a, a} < \epsilon$
By definition of the open $\epsilon$-ball of $a$ $\map {B_\epsilon} a$ in $M$:
:$a \in \map {B_\epsilon} a$
{{qed}}
Category:Open Balls
Category:Center is Element of Open Ball
mhedjg54kx16zaahve7kuvtjszzt... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Positive Real Number|positive real number]].
Let $\map {B_\epsilon} a$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball of $a$]] in $M$.
Then:
:$a \in \map {B_\epsil... | By {{Metric-space-axiom|1}}:
:$\map d {a, a} = 0$
By assumption:
:$\epsilon > 0$
Hence:
:$\map d {a, a} < \epsilon$
By definition of the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball of $a$]] $\map {B_\epsilon} a$ in $M$:
:$a \in \map {B_\epsilon} a$
{{qed}}
[[Category:Open Balls]]
[[Category:Center i... | Center is Element of Open Ball | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball | [
"Open Balls",
"Center is Element of Open Ball"
] | [
"Definition:Metric Space",
"Definition:Positive/Real Number",
"Definition:Open Ball"
] | [
"Definition:Open Ball",
"Category:Open Balls",
"Category:Center is Element of Open Ball"
] |
proofwiki-16474 | Center is Element of Open Ball/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$.
Then:
:$a \in \map {B_\epsilon} a$ | Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
From Open Ball in Normed Division Ring is Open Ball in Induced Metric, $\map {B_\epsilon} a$ is the open $\epsilon$-ball of $a$ in the metric space $\struct{R,d}$.
From Center is Element of Open Ball:
:$a \in \map {B_\epsilon} a$
{{qed}}
Category:Normed Div... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map {B_\epsilon} a$ be the [[Definition:Open Ball of Normed Division Ring|open $\epsilo... | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
From [[Open Ball in Normed Division Ring is Open Ball in Induced Metric]], $\map {B_\epsilon} a$ is the [[Definition:Open Ball|open $\epsilon$-ball of $a$]] in the ... | Center is Element of Open Ball/Normed Division Ring | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball/Normed_Division_Ring | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball/Normed_Division_Ring | [
"Normed Division Rings",
"Center is Element of Open Ball"
] | [
"Definition:Normed Division Ring",
"Definition:Strictly Positive/Real Number",
"Definition:Open Ball/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Open Ball in Normed Division Ring is Open Ball in Induced Metric",
"Definition:Open Ball",
"Definition:Metric Space",
"Center is Element of Open Ball",
"Category:Normed Division Rings",
"Category:Center is Element ... |
proofwiki-16475 | Center is Element of Open Ball/P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
Then:
:$a \in \map {B_\epsilon} a$ | By definition, $\map {B_\epsilon} a$ is the open $\epsilon$-ball of $a$ in the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
From Center is Element of Open Ball in Normed Division Ring:
$a \in \map {B_\epsilon} a$
{{qed}}
Category:Topology of P-adic Numbers
Category:Center is Element of Open Ball
5lnu6qo7... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map {B_\epsilon}... | By definition, $\map {B_\epsilon} a$ is the [[Definition:Open Ball of Normed Division Ring|open $\epsilon$-ball of $a$]] in the [[Definition:Normed Division Ring|normed division ring]] $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
From [[Center is Element of Open Ball in Normed Division Ring]]:
$a \in \map {B_\epsilon} a$
{{... | Center is Element of Open Ball/P-adic Numbers | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball/P-adic_Numbers | https://proofwiki.org/wiki/Center_is_Element_of_Open_Ball/P-adic_Numbers | [
"Topology of P-adic Numbers",
"Center is Element of Open Ball"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Open Ball/P-adic Numbers"
] | [
"Definition:Open Ball/Normed Division Ring",
"Definition:Normed Division Ring",
"Center is Element of Open Ball/Normed Division Ring",
"Category:Topology of P-adic Numbers",
"Category:Center is Element of Open Ball"
] |
proofwiki-16476 | Equivalence of Definitions of Non-Archimedean Division Ring Norm | Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.
{{TFAE|def = Non-Archimedean Division Ring Norm}}
=== Definition 1 ===
{{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}}
=== Definition 2 ===
{{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 2}} | === Definition 1 implies Definition 2 ===
Let $\norm {\,\cdot\,} : R \to \R_{\ge 0}$ be a norm on a division ring satisfying:
{{begin-axiom}}
{{axiom | n = \text N 4
| lc= Ultrametric Inequality:
| q = \forall x, y \in R
| ml= \norm {x + y}
| mo= \le
| mr= \max \set {\norm x, \no... | Let $\struct {R, +, \circ}$ be a [[Definition:Division Ring|division ring]] whose [[Definition:Ring Zero|zero]] is denoted $0_R$.
{{TFAE|def = Non-Archimedean Division Ring Norm}}
=== [[Definition:Non-Archimedean/Norm (Division Ring)/Definition 1|Definition 1]] ===
{{:Definition:Non-Archimedean/Norm (Division Ring)/D... | === Definition 1 implies Definition 2 ===
Let $\norm {\,\cdot\,} : R \to \R_{\ge 0}$ be a [[Definition:Norm on Division Ring|norm on a division ring]] satisfying:
{{begin-axiom}}
{{axiom | n = \text N 4
| lc= Ultrametric Inequality:
| q = \forall x, y \in R
| ml= \norm {x + y}
| mo= \l... | Equivalence of Definitions of Non-Archimedean Division Ring Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Non-Archimedean_Division_Ring_Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Non-Archimedean_Division_Ring_Norm | [
"Norm Theory"
] | [
"Definition:Division Ring",
"Definition:Ring Zero",
"Definition:Non-Archimedean/Norm (Division Ring)/Definition 1",
"Definition:Non-Archimedean/Norm (Division Ring)/Definition 2"
] | [
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring"
] |
proofwiki-16477 | Angle of Spherical Triangle from Sides | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos A = \cosec b \cosec c \paren {\cos a - \cos b \cos c}$ | {{begin-eqn}}
{{eqn | l = \cos b \cos c + \sin b \sin c \cos A
| r = \cos a
| c = Spherical Law of Cosines
}}
{{eqn | ll= \leadsto
| l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c =
}}
{{eqn | ll= \leadsto
| l = \cos A
| r = \dfrac {\cos a - \cos b \cos c} {\sin b \... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | {{begin-eqn}}
{{eqn | l = \cos b \cos c + \sin b \sin c \cos A
| r = \cos a
| c = [[Spherical Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c =
}}
{{eqn | ll= \leadsto
| l = \cos A
| r = \dfrac {\cos a - \cos b \cos c} {\sin... | Angle of Spherical Triangle from Sides | https://proofwiki.org/wiki/Angle_of_Spherical_Triangle_from_Sides | https://proofwiki.org/wiki/Angle_of_Spherical_Triangle_from_Sides | [
"Spherical Trigonometry"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"Spherical Law of Cosines"
] |
proofwiki-16478 | Sine of Half Angle for Spherical Triangles | :$\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b \cos c + \sin b \sin c \paren {1 - 2 \sin^2 \dfrac A 2}
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | r = \map \cos {b - c} - 2 \sin b \sin c \sin^2 \dfra... | :$\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b \cos c + \sin b \sin c \paren {1 - 2 \sin^2 \dfrac A 2}
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | r = \map \cos {b - c} - 2 \sin b \sin c \sin^2 \... | Sine of Half Angle for Spherical Triangles | https://proofwiki.org/wiki/Sine_of_Half_Angle_for_Spherical_Triangles | https://proofwiki.org/wiki/Sine_of_Half_Angle_for_Spherical_Triangles | [
"Half Angle Formulas for Spherical Triangles"
] | [] | [
"Spherical Law of Cosines",
"Cosine of Difference",
"Prosthaphaeresis Formulas/Cosine minus Cosine"
] |
proofwiki-16479 | Open Ball in P-adic Numbers is Closed Ball | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$
:Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$.
Then:
:$\forall n \in \Z : \map {B_{p... | Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x \in \map { B_{p^{-n} } } a
| o = \leadstoandfrom
| r = \norm {x - a}_p < p^{-n}
| c = {{Defof|Open Ball in P-adic Numbers|Open Ball in $p$-adic Numbers}}
}}
{{eqn | o = \leadstoandfrom
| r = \norm {x - a}_p \le p^{-\paren {n + 1} }
| c = $p... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
:Let $\map {B_\epsilon} a$ denote the [[Definition:Open Ball in P-adic Numbers|open $\epsilon$-bal... | Let $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x \in \map { B_{p^{-n} } } a
| o = \leadstoandfrom
| r = \norm {x - a}_p < p^{-n}
| c = {{Defof|Open Ball in P-adic Numbers|Open Ball in $p$-adic Numbers}}
}}
{{eqn | o = \leadstoandfrom
| r = \norm {x - a}_p \le p^{-\paren {n + 1} }
| c = [... | Open Ball in P-adic Numbers is Closed Ball | https://proofwiki.org/wiki/Open_Ball_in_P-adic_Numbers_is_Closed_Ball | https://proofwiki.org/wiki/Open_Ball_in_P-adic_Numbers_is_Closed_Ball | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers"
] | [
"P-adic Norm of p-adic Number is Power of p",
"Definition:Set Equality",
"Category:Topology of P-adic Numbers"
] |
proofwiki-16480 | Tangent of Half Angle for Spherical Triangles | :$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \tan \dfrac A 2
| r = \dfrac {\sqrt {\dfrac {\sin \paren {s - b} \sin \paren {s - c} } {\sin b \sin c} } } {\sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} } }
| c = Sine of Half Angle for Spherical Triangles, Cosine of Half Angle for Spherical Triangles
}}
{{eqn | r =... | :$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \tan \dfrac A 2
| r = \dfrac {\sqrt {\dfrac {\sin \paren {s - b} \sin \paren {s - c} } {\sin b \sin c} } } {\sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} } }
| c = [[Sine of Half Angle for Spherical Triangles]], [[Cosine of Half Angle for Spherical Triangles]]
}}
{{e... | Tangent of Half Angle for Spherical Triangles | https://proofwiki.org/wiki/Tangent_of_Half_Angle_for_Spherical_Triangles | https://proofwiki.org/wiki/Tangent_of_Half_Angle_for_Spherical_Triangles | [
"Half Angle Formulas for Spherical Triangles"
] | [] | [
"Sine of Half Angle for Spherical Triangles",
"Cosine of Half Angle for Spherical Triangles"
] |
proofwiki-16481 | Countable Basis for P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.
Then:
:$\BB_p = \set {\m... | From Sequence of Powers of Number less than One, $\sequence{p^{-n}}$ is a real null sequence.
From Null Sequence induces Local Basis in Metric Space, for all $a \in \Q_p$ the set $\set {\map {B_{p^{-n} } } a : n \in \Z}$ is a local basis of $a$.
From Union of Local Bases is Basis, the set:
:$\BB' = \ds \bigcup_{a \in \... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | From [[Sequence of Powers of Number less than One]], $\sequence{p^{-n}}$ is a [[Definition:Real Null Sequence|real null sequence]].
From [[Null Sequence induces Local Basis in Metric Space]], for all $a \in \Q_p$ the [[Definition:Set|set]] $\set {\map {B_{p^{-n} } } a : n \in \Z}$ is a [[Definition:Local Basis|local b... | Countable Basis for P-adic Numbers | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Open Ball/P-adic Numbers",
"Definition:Countable Basis"
] | [
"Sequence of Powers of Number less than One",
"Definition:Null Sequence/Real Numbers",
"Null Sequence induces Local Basis in Metric Space",
"Definition:Set",
"Definition:Local Basis",
"Union of Local Bases is Basis",
"Definition:Set",
"Definition:Basis (Topology)",
"Rational Numbers are Dense Subfie... |
proofwiki-16482 | P-adic Numbers is Second Countable Topological Space | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is second-countable. | From Countable Basis for P-adic Numbers, the topological space $\struct {\Q_p, \tau_p}$ has a countable basis.
By definition, $\struct {\Q_p, \tau_p}$ is second-countable.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | From [[Countable Basis for P-adic Numbers]], the [[Definition:Topological Space|topological space]] $\struct {\Q_p, \tau_p}$ has a [[Definition:Countable Basis|countable basis]].
By definition, $\struct {\Q_p, \tau_p}$ is [[Definition:Second-Countable Space|second-countable]].
{{qed}} | P-adic Numbers is Second Countable Topological Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Second_Countable_Topological_Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Second_Countable_Topological_Space | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topological Space",
"Definition:Second-Countable Space"
] | [
"Countable Basis for P-adic Numbers",
"Definition:Topological Space",
"Definition:Countable Basis",
"Definition:Second-Countable Space"
] |
proofwiki-16483 | P-adic Numbers is Totally Disconnected Topological Space | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is totally disconnected. | From P-adic Numbers form Non-Archimedean Valued Field:
:$\norm {\,\cdot\,}_p$ is a non-Archimedean norm.
From Non-Archimedean Division Ring is Totally Disconnected:
:$\struct {\Q_p, \tau_p}$ is totally disconnected.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:$\norm {\,\cdot\,}_p$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]].
From [[Non-Archimedean Division Ring is Totally Disconnected]]:
:$\struct {\Q_p, \tau_p}$ is [[Definition:Totally Disconnected Space|totally disconnected]].
{{q... | P-adic Numbers is Totally Disconnected Topological Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Totally_Disconnected_Topological_Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Totally_Disconnected_Topological_Space | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topological Space",
"Definition:Totally Disconnected Space"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Non-Archimedean Division Ring is Totally Disconnected",
"Definition:Totally Disconnected Space"
] |
proofwiki-16484 | P-adic Numbers is Hausdorff Topological Space | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff. | Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$.
By definition of the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$, $\tau_p$ is the topology induced by the metric $d_p$.
From Metric Space is Hausdorff, it follows that $\struct{\Q_p, \tau_p}$ is Hausdorff.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | Let $d_p$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}_p$.
By definition of the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdot\,}_p$, $\ta... | P-adic Numbers is Hausdorff Topological Space/Proof 1 | https://proofwiki.org/wiki/P-adic_Numbers_is_Hausdorff_Topological_Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Hausdorff_Topological_Space/Proof_1 | [
"Topology of P-adic Numbers",
"P-adic Numbers is Hausdorff Topological Space"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topological Space",
"Definition:T2 Space"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topology Induced by Metric/Definition 2",
"Metric Space is T2",
"Definition:T2 Space"
] |
proofwiki-16485 | P-adic Numbers is Hausdorff Topological Space | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff. | Let $x, y \in \Q_p$ such that $x \ne y$.
By {{NormAxiomNonArch|1}}:
:$r := \norm {x - y}_p > 0$
Then, for all $z\in\Q_p$ we have:
{{begin-eqn}}
{{eqn | r = \norm {(x-z) + (z-y)} _p
| l = \max\set {\norm {x - z}_p, \norm {z - y}_p}
| c = {{NormAxiomNonArch|4}}
| o = \ge
}}
{{eqn | r = \norm {x - y} _p
... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | Let $x, y \in \Q_p$ such that $x \ne y$.
By {{NormAxiomNonArch|1}}:
:$r := \norm {x - y}_p > 0$
Then, for all $z\in\Q_p$ we have:
{{begin-eqn}}
{{eqn | r = \norm {(x-z) + (z-y)} _p
| l = \max\set {\norm {x - z}_p, \norm {z - y}_p}
| c = {{NormAxiomNonArch|4}}
| o = \ge
}}
{{eqn | r = \norm {x - y} _... | P-adic Numbers is Hausdorff Topological Space/Proof 2 | https://proofwiki.org/wiki/P-adic_Numbers_is_Hausdorff_Topological_Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Hausdorff_Topological_Space/Proof_2 | [
"Topology of P-adic Numbers",
"P-adic Numbers is Hausdorff Topological Space"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topological Space",
"Definition:T2 Space"
] | [
"Definition:Open Ball/P-adic Numbers",
"Definition:Disjoint Sets"
] |
proofwiki-16486 | P-adic Numbers is Locally Compact Topological Space | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is locally compact. | From Local Basis of P-adic Number:
:$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of $a$.
From Open and Closed Balls in P-adic Numbers are Compact Subspaces;
:$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of compact sets of $a$.
Hence $\s... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | From [[Local Basis of P-adic Number]]:
:$\forall a \in \Q_p$: the [[Definition:Set|set]] $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a [[Definition:Local Basis|local basis]] of $a$.
From [[Open and Closed Balls in P-adic Numbers are Compact Subspaces]];
:$\forall a \in \Q_p$: the [[Definition:Set|set]] $\set {\map {B_{... | P-adic Numbers is Locally Compact Topological Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Locally_Compact_Topological_Space | https://proofwiki.org/wiki/P-adic_Numbers_is_Locally_Compact_Topological_Space | [
"Topology of P-adic Numbers",
"Locally Compact Spaces"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topological Space",
"Definition:Locally Compact Space"
] | [
"Local Basis of P-adic Number",
"Definition:Set",
"Definition:Local Basis",
"Open and Closed Balls in P-adic Numbers are Compact Subspaces",
"Definition:Set",
"Definition:Local Basis",
"Definition:Compact Topological Space",
"Definition:Locally Compact Space"
] |
proofwiki-16487 | Open and Closed Balls in P-adic Numbers are Compact Subspaces | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n}}} a$ are compact. | We begin by proving the theorem for the closed ball $\map {B^-_{p^{-n} } } a$.
From Open Ball in P-adic Numbers is Closed Ball then the theorem will be proved.
Let $d$ denote the subspace metric induced on $\map {B^-_{p^{-n}}} a$ by the $p$-adic Metric.
From Open and Closed Balls in P-adic Numbers are Totally Bounded:
... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the [[Definition:Open Ball in P-adic Numbers|open ball]] $\map {B_{p^{-n} } } a$ and [[Definition:Closed Bal... | We begin by proving the [[Definition:Theorem|theorem]] for the [[Definition:Closed Ball in P-adic Numbers|closed ball]] $\map {B^-_{p^{-n} } } a$.
From [[Open Ball in P-adic Numbers is Closed Ball]] then the [[Definition:Theorem|theorem]] will be proved.
Let $d$ denote the [[Definition:Metric Subspace|subspace metri... | Open and Closed Balls in P-adic Numbers are Compact Subspaces | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Compact_Subspaces | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Compact_Subspaces | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Compact Space/Metric Space"
] | [
"Definition:Theorem",
"Definition:Closed Ball/P-adic Numbers",
"Open Ball in P-adic Numbers is Closed Ball",
"Definition:Theorem",
"Definition:Metric Subspace",
"Definition:P-adic Metric",
"Open and Closed Balls in P-adic Numbers are Totally Bounded",
"Definition:Closed Ball/P-adic Numbers",
"Defini... |
proofwiki-16488 | Ambiguous Case for Triangle Side-Side-Angle Congruence | Let $\triangle ABC$ be a triangle.
Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively.
Let the sides $a$ and $b$ be known.
Let the angle $\angle B$ also be known.
Then it may not be possible to know the value of $\angle A$.
This is known as the '''ambiguous case'''. | From the Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin B} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is not possibl... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let the [[Definition:Side of Polygon|sides]] $a, b, c$ of $\triangle ABC$ be [[Definition:Opposite (in Triangle)|opposite]] $A, B, C$ respectively.
Let the [[Definition:Side of Polygon|sides]] $a$ and $b$ be known.
Let the [[Definition:Angle|angl... | From the [[Law of Sines]], we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin B} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is no... | Ambiguous Case for Triangle Side-Side-Angle Congruence/Proof 1 | https://proofwiki.org/wiki/Ambiguous_Case_for_Triangle_Side-Side-Angle_Congruence | https://proofwiki.org/wiki/Ambiguous_Case_for_Triangle_Side-Side-Angle_Congruence/Proof_1 | [
"Ambiguous Case for Triangle Side-Side-Angle Congruence",
"Ambiguous Case",
"Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Side",
"Definition:Angle",
"Ambiguous Case for Triangle Side-Side-Angle Congruence"
] | [
"Law of Sines",
"File:Ambiguous-Case.png"
] |
proofwiki-16489 | Analogue Formula for Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
{{begin-eqn}}
{{eqn | l = \sin a \cos B
| r = \cos b \sin c - \sin b \cos c ... | {{begin-eqn}}
{{eqn | l = \sin c \sin a \cos B
| r = \cos b - \cos c \cos a
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b - \cos c \paren {\cos b \cos c + \sin b \sin c \cos A}
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b \paren {1 - \cos^2 c} - \sin b \sin c \cos c \cos A
| c = r... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | {{begin-eqn}}
{{eqn | l = \sin c \sin a \cos B
| r = \cos b - \cos c \cos a
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b - \cos c \paren {\cos b \cos c + \sin b \sin c \cos A}
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b \paren {1 - \cos^2 c} - \sin b \sin c \cos c \cos A
... | Analogue Formula for Spherical Law of Cosines/Proof 1 | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_1 | [
"Spherical Trigonometry",
"Spherical Law of Cosines",
"Analogue Formula for Spherical Law of Cosines"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"Spherical Law of Cosines",
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine",
"Spherical Law of Cosines",
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-16490 | Analogue Formula for Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
{{begin-eqn}}
{{eqn | l = \sin a \cos B
| r = \cos b \sin c - \sin b \cos c ... | :500px
Suppose $c$ is less than $\dfrac \pi 2$.
Let $BA$ be produced to $D$ so that $BD = \dfrac \pi 2$.
Then:
:$AD = \dfrac \pi 2 - c$
and:
:$\angle CAD = pi - A$
Let $C$ and $D$ be joined by an arc of a great circle, denoted $x$.
From the triangle $\sphericalangle DAC$, using the Spherical Law of Cosines:
{{begin-eqn... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-Analog.png|500px]]
Suppose $c$ is less than $\dfrac \pi 2$.
Let $BA$ be [[Definition:Production|produced]] to $D$ so that $BD = \dfrac \pi 2$.
Then:
:$AD = \dfrac \pi 2 - c$
and:
:$\angle CAD = pi - A$
Let $C$ and $D$ be joined by an [[Definition:Arc of Circle|arc]] of a [[Definitio... | Analogue Formula for Spherical Law of Cosines/Proof 2 | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_2 | [
"Spherical Trigonometry",
"Spherical Law of Cosines",
"Analogue Formula for Spherical Law of Cosines"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-Analog.png",
"Definition:Production",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Spherical Triangle",
"Spherical Law of Cosines",
"Definition:Spherical Triangle",
"Spherical Law of Cosines",
"Definition:Point"
] |
proofwiki-16491 | Analogue Formula for Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
{{begin-eqn}}
{{eqn | l = \sin a \cos B
| r = \cos b \sin c - \sin b \cos c ... | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $O$ be joined to each of $A$, $B$ and $C... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-2.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$,... | Analogue Formula for Spherical Law of Cosines/Proof 3 | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_3 | [
"Spherical Trigonometry",
"Spherical Law of Cosines",
"Analogue Formula for Spherical Law of Cosines"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-2.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-16492 | Four-Parts Formula | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
We have:
:$\cos a \cos C = \sin a \cot b - \sin C \cot B$
That is:
:$\map \cos {\text {i... | :400px
{{begin-eqn}}
{{eqn | l = \cos b
| r = \cos a \cos c + \sin a \sin c \cos B
| c = Spherical Law of Cosines
}}
{{eqn | l = \cos c
| r = \cos b \cos a + \sin b \sin a \cos C
| c = Spherical Law of Cosines
}}
{{eqn | ll= \leadsto
| l = \cos b
| r = \cos a \paren {\cos b \cos a + ... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:SphericalTriangle-FourParts.png|400px]]
{{begin-eqn}}
{{eqn | l = \cos b
| r = \cos a \cos c + \sin a \sin c \cos B
| c = [[Spherical Law of Cosines]]
}}
{{eqn | l = \cos c
| r = \cos b \cos a + \sin b \sin a \cos C
| c = [[Spherical Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = ... | Four-Parts Formula | https://proofwiki.org/wiki/Four-Parts_Formula | https://proofwiki.org/wiki/Four-Parts_Formula | [
"Spherical Trigonometry"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:SphericalTriangle-FourParts.png",
"Spherical Law of Cosines",
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine",
"Spherical Law of Sines"
] |
proofwiki-16493 | Countable Basis for P-adic Numbers/Closed Balls | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.
Then:
:$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$
is a countable basis for $\struct{\Q_p, \tau_p}$. | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.
From Open Ball in P-adic Numbers is Closed Ball:
:$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z} = \set {\map {B_{p^{-n + 1} } } q : q \in \Q, n \in \Z} = \set {\map {B_{p^{-n} } } q : q \in \... | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed $\epsilon$-ball]] of $a$.
Then:
:$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$
is a [[Definition:Countable Basis|countable basis]] for $\struct{\Q_p, \tau_p}$. | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the [[Definition:Open Ball in P-adic Numbers|open $\epsilon$-ball]] of $a$.
From [[Open Ball in P-adic Numbers is Closed Ball]]:
:$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z} = \set {\map {B_{p^{-n + 1} } } q : q \in \Q, ... | Countable Basis for P-adic Numbers/Closed Balls | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers/Closed_Balls | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers/Closed_Balls | [
"Topology of P-adic Numbers"
] | [
"Definition:Closed Ball/P-adic Numbers",
"Definition:Countable Basis"
] | [
"Definition:Open Ball/P-adic Numbers",
"Open Ball in P-adic Numbers is Closed Ball",
"Countable Basis for P-adic Numbers",
"Definition:Countable Basis"
] |
proofwiki-16494 | Countable Basis for P-adic Numbers/Cosets | Let $\Z_p$ be the $p$-adic integers.
Then:
:$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z}$
is a countable basis for $\struct{\Q_p, \tau_p}$. | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.
From Closed Balls of P-adic Number:
:$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z} = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$
From Countable Closed Ball Basis for P-adic Numbers:
:$\BB_p... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z}$
is a [[Definition:Countable Basis|countable basis]] for $\struct{\Q_p, \tau_p}$. | For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed $\epsilon$-ball]] of $a$.
From [[Closed Balls of P-adic Number]]:
:$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z} = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$
From [[C... | Countable Basis for P-adic Numbers/Cosets | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers/Cosets | https://proofwiki.org/wiki/Countable_Basis_for_P-adic_Numbers/Cosets | [
"Topology of P-adic Numbers"
] | [
"Definition:P-adic Integer",
"Definition:Countable Basis"
] | [
"Definition:Closed Ball/P-adic Numbers",
"Closed Ball of P-adic Number",
"Countable Basis for P-adic Numbers/Closed Balls",
"Definition:Countable Basis"
] |
proofwiki-16495 | Summary of Topology on P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.
Then $\struct{\Q_p, \tau_p}$ is:
:$(1): \quad$ Hausdorff
:$(2): \quad$ second-countable
:$(3): \quad$ totally disconnected
:$(4): \quad$ l... | Follows from:
:P-adic Numbers is Hausdorff Topological Space
:P-adic Numbers is Second Countable Topological Space
:P-adic Numbers is Totally Disconnected Topological Space
:P-adic Numbers is Locally Compact Topological Space
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\tau_p$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Non-Archimedean Division Ring Norm|non-... | Follows from:
:[[P-adic Numbers is Hausdorff Topological Space]]
:[[P-adic Numbers is Second Countable Topological Space]]
:[[P-adic Numbers is Totally Disconnected Topological Space]]
:[[P-adic Numbers is Locally Compact Topological Space]]
{{qed}} | Summary of Topology on P-adic Numbers | https://proofwiki.org/wiki/Summary_of_Topology_on_P-adic_Numbers | https://proofwiki.org/wiki/Summary_of_Topology_on_P-adic_Numbers | [
"P-adic Number Theory",
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:T2 Space",
"Definition:Second-Countable Space",
"Definition:Totally Disconnected Space",
"Definition:Locally Compa... | [
"P-adic Numbers is Hausdorff Topological Space",
"P-adic Numbers is Second Countable Topological Space",
"P-adic Numbers is Totally Disconnected Topological Space",
"P-adic Numbers is Locally Compact Topological Space"
] |
proofwiki-16496 | Napier's Cosine Rule for Right Spherical Triangles | :500px
Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$ such that the angle $\sphericalangle C$ is a right angle.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the remaining ... | We are given that $\sphericalangle C$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the interior of the circle above, where the symbol $\Box$ denotes a right angle.
==== $\sin a$ ====
{{begin-eqn}}
{{eqn | l = \dfrac {\sin a} {\sin A}
| r = \dfrac {\sin c} {\sin C}
| ... | :[[File:Right-spherical-triangle.png|500px]]
Let $\triangle ABC$ be a [[Definition:Right Spherical Triangle|right spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$ such that the [[Definition:Spherical Angle|angle]] $\sphericalangle C$ i... | We are given that $\sphericalangle C$ is a [[Definition:Right Angle|right angle]].
Let the remaining parts of $\triangle ABC$ be arranged according to the [[Definition:Interior of Geometric Figure|interior]] of the [[Definition:Circle|circle]] above, where the [[Definition:Symbol|symbol]] $\Box$ denotes a [[Definition... | Napier's Cosine Rule for Right Spherical Triangles | https://proofwiki.org/wiki/Napier's_Cosine_Rule_for_Right_Spherical_Triangles | https://proofwiki.org/wiki/Napier's_Cosine_Rule_for_Right_Spherical_Triangles | [
"Napier's Rules for Right Spherical Triangles"
] | [
"File:Right-spherical-triangle.png",
"Definition:Right Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Angle",
"Definition:Right Angle",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"... | [
"Definition:Right Angle",
"Definition:Interior of Geometric Figure",
"Definition:Circle",
"Definition:Symbol",
"Definition:Right Angle",
"Spherical Law of Sines",
"Definition:Spherical Triangle/Side",
"Sine of Right Angle",
"Cosine of Complement equals Sine",
"Spherical Law of Sines",
"Definitio... |
proofwiki-16497 | Napier's Tangent Rule for Right Spherical Triangles | :500px
Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$ such that the angle $\sphericalangle C$ is a right angle.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the remaining ... | We are given that $\sphericalangle C$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the interior of the circle above, where the symbol $\Box$ denotes a right angle.
==== $\sin a$ ====
{{begin-eqn}}
{{eqn | l = \cos a \cos C
| r = \sin a \cot b - \sin C \cot B
| c = Fo... | :[[File:Right-spherical-triangle.png|500px]]
Let $\triangle ABC$ be a [[Definition:Right Spherical Triangle|right spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$ such that the [[Definition:Spherical Angle|angle]] $\sphericalangle C$ i... | We are given that $\sphericalangle C$ is a [[Definition:Right Angle|right angle]].
Let the remaining parts of $\triangle ABC$ be arranged according to the [[Definition:Interior of Geometric Figure|interior]] of the [[Definition:Circle|circle]] above, where the [[Definition:Symbol|symbol]] $\Box$ denotes a [[Definition... | Napier's Tangent Rule for Right Spherical Triangles | https://proofwiki.org/wiki/Napier's_Tangent_Rule_for_Right_Spherical_Triangles | https://proofwiki.org/wiki/Napier's_Tangent_Rule_for_Right_Spherical_Triangles | [
"Napier's Rules for Right Spherical Triangles"
] | [
"File:Right-spherical-triangle.png",
"Definition:Right Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Angle",
"Definition:Right Angle",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"... | [
"Definition:Right Angle",
"Definition:Interior of Geometric Figure",
"Definition:Circle",
"Definition:Symbol",
"Definition:Right Angle",
"Four-Parts Formula",
"Cosine of Right Angle",
"Sine of Right Angle",
"Tangent of Complement equals Cotangent",
"Four-Parts Formula",
"Cosine of Right Angle",
... |
proofwiki-16498 | Napier's Cosine Rule for Quadrantal Triangles | :500px
Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the side $c$ be a right angle.
Let the remaining parts of $\triangle ABC$ be... | We are given that $c$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of the circle above, where the symbol $\Box$ denotes a right angle.
==== $\sin A$ ====
{{begin-eqn}}
{{eqn | l = \dfrac {\sin A} {\sin a}
| r = \dfrac {\sin C} {\sin c}
| c = Spheri... | :[[File:Quadrantal-spherical-triangle.png|500px]]
Let $\triangle ABC$ be a [[Definition:Quadrantal Triangle|quadrantal triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\tria... | We are given that $c$ is a [[Definition:Right Angle|right angle]].
Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of the [[Definition:Circle|circle]] above, where the [[Definition:Symbol|symbol]] $\Box$ denotes a [[Definition:Right Angle|right angle]].
==== $\sin A$ ====
{{be... | Napier's Cosine Rule for Quadrantal Triangles | https://proofwiki.org/wiki/Napier's_Cosine_Rule_for_Quadrantal_Triangles | https://proofwiki.org/wiki/Napier's_Cosine_Rule_for_Quadrantal_Triangles | [
"Napier's Rules for Quadrantal Triangles"
] | [
"File:Quadrantal-spherical-triangle.png",
"Definition:Quadrantal Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Spherical Triangle/Side",
"Defin... | [
"Definition:Right Angle",
"Definition:Circle",
"Definition:Symbol",
"Definition:Right Angle",
"Spherical Law of Sines",
"Definition:Spherical Angle",
"Sine of Right Angle",
"Cosine of Complement equals Sine",
"Cosine Function is Even",
"Spherical Law of Sines",
"Definition:Spherical Angle",
"S... |
proofwiki-16499 | Napier's Tangent Rule for Quadrantal Triangles | :500px
Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the side $c$ be a right angle.
Let the remaining parts of $\triangle ABC$ be... | We are given that $c$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of the circle above, where the symbol $\Box$ denotes a right angle.
==== $\sin A$ ====
{{begin-eqn}}
{{eqn | l = \cos A \cos c
| r = \sin A \cot B - \sin c \cot b
| c = Four-Parts F... | :[[File:Quadrantal-spherical-triangle.png|500px]]
Let $\triangle ABC$ be a [[Definition:Quadrantal Triangle|quadrantal triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\tria... | We are given that $c$ is a [[Definition:Right Angle|right angle]].
Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of the [[Definition:Circle|circle]] above, where the [[Definition:Symbol|symbol]] $\Box$ denotes a [[Definition:Right Angle|right angle]].
==== $\sin A$ ====
{{be... | Napier's Tangent Rule for Quadrantal Triangles | https://proofwiki.org/wiki/Napier's_Tangent_Rule_for_Quadrantal_Triangles | https://proofwiki.org/wiki/Napier's_Tangent_Rule_for_Quadrantal_Triangles | [
"Napier's Rules for Quadrantal Triangles"
] | [
"File:Quadrantal-spherical-triangle.png",
"Definition:Quadrantal Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Spherical Triangle/Side",
"Defin... | [
"Definition:Right Angle",
"Definition:Circle",
"Definition:Symbol",
"Definition:Right Angle",
"Four-Parts Formula",
"Cosine of Right Angle",
"Sine of Right Angle",
"Tangent of Complement equals Cotangent",
"Four-Parts Formula",
"Cosine of Right Angle",
"Sine of Right Angle",
"Tangent of Comple... |
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