avewright/proofwiki-math huggingface dataset:
Theorem: 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:
:$0 \le \log_{10} a < 1$

Proof:
{{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|Common Logarithm}}
}}
{{end-eqn}}
We are given that:
:$1 \le a < 10$
It follows from Range of Common Logarithm of Number between 1 and 10 that:
:$0 \le \log_{10} a < 1$
{{qed}}
Category:Common Logarithms
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Theorem: 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 $x$.

Proof:
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 the exponent of $x$.
By Multiplication by Power of 10 by Moving Decimal Point, multiplication by $10^e$ is the same as moving the decimal point $e$ places to the right.
Hence $a \times 10^e$ has $e$ more digits to the left of the decimal point than $a$ does.
That is: $e + 1$.
Hence the result.
{{qed}}
Theorem: 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 $x$.

Proof:
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 $\log_{10} x$ is equal to the exponent of $x$.
When $e = -1$, we have that:
:$0 \cdotp 1 \le x < 1$
and so $x$ is in the form:
:$x = 0 \cdotp rstu \cdotp$
where $rstu \ldots$ are the digits of $x$ following the decimal point such that $r > 0$.
Thus when $e = -1$ there are no zero digits to the immediate right of the decimal point of $x$
That is $e$ is one less than the number of zero digits to the immediate right of the decimal point of $x$.
By Multiplication by Power of 10 by Moving Decimal Point, multiplication by $10^{-e}$ is the same as moving the decimal point $e$ places to the right.
Hence $a \times 10^{-e}$ has $e - 1$ more digits to the left of the decimal point than $a$ does.
That is: $e - 1$.
Hence the result.
{{qed}}
Theorem: Intersection of Plane with Sphere is Circle

The intersection of a plane with a sphere is a circle.

Proof:
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 $P$ is parallel to the plane $z = 0$.
Thus from the Equation of Plane we have that $P$ can be described as:
:$z = c$
where $\size c < R$ (or $P$ would not intersect $S$).
Let $A = \tuple {x, y, z}$ be an arbitrary point on $S \cap P$.
We have that:
{{begin-eqn}}
{{eqn | l = x^2 + y^2 + z^2
      | r = R^2
      | c = Equation of Sphere
}}
{{eqn | ll= \leadsto
      | l = x^2 + y^2 + c^2
      | r = R^2
      | c = Equation of Plane: $z - c$
}}
{{eqn | ll= \leadsto
      | l = x^2 + y^2
      | r = R^2 - c^2
      | c = 
}}
{{eqn | o = >
      | r = 0
      | c = as $-R < c < R$
}}
{{end-eqn}}
The result follows from Equation of Circle.
{{qed}}
Theorem: 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 = \forall x \in \Q_p
        | ml= \norm x_p = 0
        | mo= \iff
        | mr= x = 0
}}
{{axiom | n = \text N 2
        | lc= Multiplicativity:
        | q = \forall x, y \in \Q_p
        | ml= \norm {x \cdot y}_p
        | mo= =
        | mr= \norm x_p \times \norm y_p
}}
{{axiom | n = \text N 4
        | lc= Ultrametric Inequality:
        | q = \forall x, y \in Q_p
        | ml= \norm {x + y}_p
        | mo= \le
        | mr= \max \set {\norm x_p, \norm y_p}
}}
{{end-axiom}}

Proof:
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
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Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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 $E$, it is on either one side or the other.
Hence there is a finite spherical angle $\phi$ between $E$ and $C$.
Let a diameter $PQ$ of $S$ be constructed whose endpoints are on $E$ such that:
:neither $P$ nor $Q$ coincides with $A$ or $B$
:both $A$ and $B$ are on the same semicircle of $E$ into which $PQ$ divides $E$.
Let two great circles $F$ and $G$ be constructed through $PQ$ which are at a spherical angle $\dfrac \phi 2$ from $E$, one in one direction and one in the other.
Then $F$ and $G$ both divide $S$ into two hemispheres:
:one such division has a hemisphere which contains $A$ and $B$ but does not contain $E$
:the other such division has a hemisphere which contains $A$, $B$ and $E$.
Hence the result.
{{qed}}
Theorem: 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.

Proof:
$A$ 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 $E$.
The result follows by definition of spherical angle and length of side of $AB$.
{{qed}}
Theorem: 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:
:$a \in \map {{B_\epsilon}^-} a$

Proof:
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
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Theorem: 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:
:$a \in \map { {B_\epsilon}^-} a$

Proof:
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}}
Category:Normed Division Rings
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Theorem: 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:
:$a \in \map { {B_\epsilon}^-} a$

Proof:
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
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Theorem: 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.

Proof:
: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 to $OA$ and $OB$ respectively.
Thus:
{{begin-eqn}}
{{eqn | l = \angle CRD
      | r = \angle AOB
      | c = 
}}
{{eqn | ll= \leadsto
      | l = CD
      | r = \dfrac {RC} {OA} AB
      | c = 
}}
{{eqn | r = \dfrac {RC} {OC} AB
      | c = as $OA = OC$ are both radii of $S$
}}
{{end-eqn}}
We also have that:
{{begin-eqn}}
{{eqn | l = RC
      | o = \perp
      | r = OR
      | c = 
}}
{{eqn | ll= \leadsto
      | l = RC
      | r = OC \cos \angle RCO
      | c = 
}}
{{end-eqn}}
and that:
{{begin-eqn}}
{{eqn | l = RC
      | o = \parallel
      | r = OA
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \angle RCO
      | r = \angle AOC
      | c = 
}}
{{end-eqn}}
We have that $\angle AOC$ is the (plane) angle subtended at $O$ by the arc $AC$ of $\bigcirc EAB$.
Thus:
{{begin-eqn}}
{{eqn | l = CD
      | r = AB \cos AC
      | c = 
}}
{{eqn | r = AB \, \map \cos {PA - PC}
      | c = 
}}
{{eqn | r = AB \sin PC
      | c = as $PA$ is a right angle, and Cosine of Complement equals Sine
}}
{{end-eqn}}
Hence the result.
{{qed}}
Theorem: 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 $\operatorname {Long}_K$ denote the longitude of $J$ and $K$ respectively, measured in degrees.
Let $\operatorname {Lat}_J$ denote the latitude of $J$ (and $K$).
Then: 
:$JK \approx 60 \times \size {\operatorname {Long}_J - \operatorname {Long}_K} \cos \operatorname {Lat}_J$ nautical miles

Proof:
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 definition of nautical mile:
:$1$ nautical mile $= \dfrac 1 {60}$ degree of arc of latitude along a meridian measured along Earth's surface.
Thus, approximately, $1$ degree of arc along the equator equals $60$ nautical miles.
The result follows.
{{qed}}
Theorem: 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:
:$a \in \map {B_\epsilon} a$

Proof:
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
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Theorem: 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:
:$a \in \map {B_\epsilon} a$

Proof:
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 Division Rings
Category:Center is Element of Open Ball
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Theorem: 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:
:$a \in \map {B_\epsilon} a$

Proof:
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
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Theorem: 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}}

Proof:
=== 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, \norm y}
}}
{{end-axiom}}
It remains only to show that $\norm {\,\cdot\,}$ satisfies $(\text N 1)$ and $(\text N 2)$.
This follows from the definition of a norm on a division ring.
{{qed|lemma}}
Theorem: 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:
:$\cos A = \cosec b \cosec c \paren {\cos a - \cos b \cos c}$

Proof:
{{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 \sin c}
      | c = 
}}
{{eqn | r = \cosec b \cosec c \paren {\cos a - \cos b \cos c}
      | c = {{Defof|Real Cosecant Function}}
}}
{{end-eqn}}
{{qed}}
Theorem: 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 $s = \dfrac {a + b + c} 2$.

Proof:
{{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 \dfrac A 2
      | c = Cosine of Difference
}}
{{eqn | ll= \leadsto
      | l = \map \cos {b - c} - \cos a
      | r = 2 \sin b \sin c \sin^2 \dfrac A 2
      | c = rearranging
}}
{{eqn | ll= \leadsto
      | l = 2 \sin \dfrac {a + \paren {b - c} } 2 \sin \dfrac {a - \paren {b - c} } 2
      | r = 2 \sin b \sin c \sin^2 \dfrac A 2
      | c = Cosine minus Cosine
}}
{{eqn | ll= \leadsto
      | l = 2 \, \map \sin {\dfrac {a + b + c} 2 - c}  \, \map \sin {\dfrac {a + b + c} 2 - b}
      | r = 2 \sin b \sin c \sin^2 \dfrac A 2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \sin {s - c} \, \map \sin {s - b}
      | r = \sin b \sin c \sin^2 \dfrac A 2
      | c = setting $s = \dfrac {a + b + c} 2$ and simplifying
}}
{{end-eqn}}
The result follows.
{{qed}}
Theorem: 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:
:$\forall n \in \Z : \map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$

Proof:
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 = $p$-adic Norm of $p$-adic Number is Power of $p$
}}
{{eqn | o = \leadstoandfrom
      | r = x \in \map {B^-_{p^{-\paren {n + 1} } } } a
      | c = {{Defof|Closed Ball in P-adic Numbers|Closed Ball in $p$-adic Numbers}}
}}
{{end-eqn}}
By set equality:
:$\map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$
{{qed}}
Category:Topology of P-adic Numbers
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Theorem: 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 $s = \dfrac {a + b + c} 2$.

Proof:
{{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 = \dfrac {\sqrt {\sin \paren {s - b} \sin \paren {s - c} } } {\sqrt {\sin s \, \map \sin {s - a} } }
      | c = simplification
}}
{{end-eqn}}
The result follows.
{{qed}}
Theorem: 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:
:$\BB_p = \set {\map {B_{p^{-n} } } q : q \in \Q, n \in \Z}$
is a countable basis for $\struct{\Q_p, \tau_p}$.

Proof:
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:
:$\BB' = \ds \bigcup_{a \in \Q_p} \set {\map {B_{p^{-n} } } a : n \in \Z} = \set {\map {B_{p^{-n} } } a : a \in Q_p, n \in \Z}$
is a basis for $\tau_p$.
Let $a \in \Q_p$ and $n \in \Z$.
From Rational Numbers are Dense Subfield of P-adic Numbers:
:$\Q$ is everywhere dense in $\Q_p$.
Hence:
:$\exists q \in \Q: q \in \map {B_{p^{-n} } } a$
From Characterization of Open Ball in P-adic Numbers,
:$\map {B_{p^{-n} } } q = \map {B_{p^{-n} } } a$
Hence:
:$\map {B_{p^{-n} } } a \in \BB_p$
Since $q$ and $n$ were arbitrary, then:
:$\BB' \subseteq \BB_p$
It follows that $\BB_p$ is a basis for $\tau_p$.
It remains to show that $\BB_p$ is countable.
For all $q \in \Q$, let $\map {\BB_p} q = \set {\map {B_{p^{-n} } } q : n \in \Z}$.
Then: 
:$\BB_p = \ds \bigcup_{q \mathop \in \Q} \map {\BB_p} q$
From Integers are Countably Infinite and Surjection from Countably Infinite Set iff Countable:
:$\forall q \in \Q: \map {\BB_p} q$ is a countable set.
From Rational Numbers are Countably Infinite:
:$\BB_p$ is a countable union of countable sets.
From Countable Union of Countable Sets is Countable:
:$\BB_p$ is countable.
{{qed}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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
}}
{{eqn | r = r
}}
{{end-eqn}}
Therefore, the $r$-open balls $\map {B_r} x$ and $\map {B_r} y$ are disjoint.
{{qed}}