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-19900 | Divisor of Integer/Examples/2 divides n(n+1) | Let $n$ be an integer.
Then:
:$2 \divides n \paren {n + 1}$ | Suppose $n$ is even.
Then $2 \divides n$ by definition.
Hence from Divisor Divides Multiple:
:$2 \divides n \paren {n + 1}$
{{qed|lemma}}
Suppose $n$ is odd.
Then $n + 1$ is even
Then $2 \divides n + 1$ by definition.
Hence from Divisor Divides Multiple:
:$2 \divides n \paren {n + 1}$
{{qed}} | Let $n$ be an [[Definition:Integer|integer]].
Then:
:$2 \divides n \paren {n + 1}$ | Suppose $n$ is [[Definition:Even Integer|even]].
Then $2 \divides n$ by definition.
Hence from [[Divisor Divides Multiple]]:
:$2 \divides n \paren {n + 1}$
{{qed|lemma}}
Suppose $n$ is [[Definition:Odd Integer|odd]].
Then $n + 1$ is [[Definition:Even Integer|even]]
Then $2 \divides n + 1$ by definition.
Hence fr... | Divisor of Integer/Examples/2 divides n(n+1) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/2_divides_n(n+1) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/2_divides_n(n+1) | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Definition:Even Integer",
"Divisor Divides Multiple",
"Definition:Odd Integer",
"Definition:Even Integer",
"Divisor Divides Multiple"
] |
proofwiki-19901 | Divisor of Integer/Examples/3 divides n(n+1)(n+2) | Let $n$ be an integer.
Then:
:$3 \divides n \paren {n + 1} \paren {n + 2}$ | $n$ is of one of these forms:
{{begin-eqn}}
{{eqn | l = n
| r = 3 k
}}
{{eqn | l = n
| r = 3 k + 1
}}
{{eqn | l = n
| r = 3 k + 2
}}
{{end-eqn}}
for some $k \in \Z$.
Suppose $n = 3 k$.
Then $3 \divides n$ by definition.
Suppose $n = 3 k + 1$.
Then:
:$n + 2 = 3 k + 3 = 3 \paren {k + 1}$
Thus:
:$3 \divi... | Let $n$ be an [[Definition:Integer|integer]].
Then:
:$3 \divides n \paren {n + 1} \paren {n + 2}$ | $n$ is of one of these forms:
{{begin-eqn}}
{{eqn | l = n
| r = 3 k
}}
{{eqn | l = n
| r = 3 k + 1
}}
{{eqn | l = n
| r = 3 k + 2
}}
{{end-eqn}}
for some $k \in \Z$.
Suppose $n = 3 k$.
Then $3 \divides n$ by definition.
Suppose $n = 3 k + 1$.
Then:
:$n + 2 = 3 k + 3 = 3 \paren {k + 1}$
Thus:
... | Divisor of Integer/Examples/3 divides n(n+1)(n+2) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/3_divides_n(n+1)(n+2) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/3_divides_n(n+1)(n+2) | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Divisor Divides Multiple"
] |
proofwiki-19902 | Divisor of Integer/Examples/7 divides 2^3n - 1 | Let $n$ be an integer such that $n \ge 1$.
Then:
:$7 \divides 2^{3 n} - 1$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$7 \divides 2^{3 n} - 1$ | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Then:
:$7 \divides 2^{3 n} - 1$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$7 \divides 2^{3 n} - 1$ | Divisor of Integer/Examples/7 divides 2^3n - 1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/7_divides_2^3n_-_1 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/7_divides_2^3n_-_1 | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-19903 | Divisor of Integer/Examples/8 divides 3^2n + 7 | Let $n$ be an integer such that $n \ge 1$.
Then:
:$8 \divides 3^{2 n} + 7$ | From Integer Less One divides Power Less One, we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
Hence we have the special case where $m = 3^2$:
:$8 \divides 3^{2 n} - 1$
from which it follows immediately that:
:$8 \divides 3^{2 n} - 1 + 8 = 3^{2 n} + 7$
{{qed}} | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Then:
:$8 \divides 3^{2 n} + 7$ | From [[Integer Less One divides Power Less One]], we have that:
:$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
Hence we have the special case where $m = 3^2$:
:$8 \divides 3^{2 n} - 1$
from which it follows immediately that:
:$8 \divides 3^{2 n} - 1 + 8 = 3^{2 n} + 7$
{{qed}} | Divisor of Integer/Examples/8 divides 3^2n + 7/Proof 2 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/8_divides_3^2n_+_7 | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/8_divides_3^2n_+_7/Proof_2 | [
"Divisor of Integer/Examples/8 divides 3^2n + 7",
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Integer Less One divides Power Less One"
] |
proofwiki-19904 | Complex Contour Integral as Contour Integrals | Let $f: D \to \C$ be a complex-differentiable function, where $D \subseteq \C$ is a connected domain.
Let $u, v: \R^2 \to \R$ be defined by:
:$\map f {x + i y} = \map u {x, y} + i \map v {x, y}$
Let $C$ be a contour in $D$.
Let $\phi: \R^2 \to \C$ be defined by:
:$\map \phi {x, y} = x + i y$
Then there exists a piecewi... | First, suppose that $C$ consists of one directed smooth curve $C_1$.
Let $\gamma_1 : \closedint a b \to D$ be a smooth path that is a parameterization of $C_1$.
Define $x, y: \closedint a b \to \R$ by:
:$\map {\gamma_1} t = \map x t + i \map y t$
Then:
{{begin-eqn}}
{{eqn | l = \int_C \map f z \rd z
| r = \int_a^... | Let $f: D \to \C$ be a [[Definition:Differentiable Complex Function in Open Set|complex-differentiable function]], where $D \subseteq \C$ is a [[Definition:Connected Domain (Complex Analysis)|connected domain]].
Let $u, v: \R^2 \to \R$ be defined by:
:$\map f {x + i y} = \map u {x, y} + i \map v {x, y}$
Let $C$ be ... | First, suppose that $C$ consists of one [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curve]] $C_1$.
Let $\gamma_1 : \closedint a b \to D$ be a [[Definition:Smooth Path (Complex Analysis)|smooth path]] that is a [[Definition:Parameterization of Directed Smooth Curve (Complex Plane)|parameterizatio... | Complex Contour Integral as Contour Integrals | https://proofwiki.org/wiki/Complex_Contour_Integral_as_Contour_Integrals | https://proofwiki.org/wiki/Complex_Contour_Integral_as_Contour_Integrals | [
"Complex Contour Integrals",
"Contour Integrals"
] | [
"Definition:Holomorphic Function/Complex Plane",
"Definition:Connected Domain (Complex Analysis)",
"Definition:Contour/Complex Plane",
"Definition:Piecewise Continuously Differentiable Function",
"Definition:Contour Integral/Complex",
"Definition:Contour Integral",
"Definition:Complex-Valued Function",
... | [
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Smooth Path/Complex",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Linear Combination of Integrals/Definite",
"Definition:Smooth Path/Complex",
"Definition:Continuously Differentiable/Vector-Valued Function",
"Definitio... |
proofwiki-19905 | Divisor of Integer/Examples/3 divides 2^n + (-1)^(n+1) | Let $n$ be an integer such that $n \ge 1$.
Then:
:$3 \divides 2^n + \paren {-1}^{n + 1}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$3 \divides 2^n + \paren {-1}^{n + 1}$ | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Then:
:$3 \divides 2^n + \paren {-1}^{n + 1}$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$3 \divides 2^n + \paren {-1}^{n + 1}$ | Divisor of Integer/Examples/3 divides 2^n + (-1)^(n+1) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/3_divides_2^n_+_(-1)^(n+1) | https://proofwiki.org/wiki/Divisor_of_Integer/Examples/3_divides_2^n_+_(-1)^(n+1) | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-19906 | Chinese Remainder Theorem (Groups) | Let $G$ be a group.
Let $N_1, \ldots, N_n$ for some $n \ge 1$ be normal subgroups of $G$.
Let $\pi_i: G \rightarrow G / N_i$ be the canonical projections.
Then the homomorphism $\pi: G \to G / N_1 \times \cdots \times G / N_n$ defined as:
:$\map \pi x = \tuple {\map {\pi_1} x, \ldots, \map {\pi_n} x}$
has the kernel $\... | By Quotient Group Epimorphism is Epimorphism, each canonical projection $\pi_i: G \to G / N_i$ is a group homomorphism.
Hence the mapping $\pi$ is indeed a group homomorphism.
The kernel of $\pi$ is given by:
:$\ds \ker \pi = \set {x \in G: \forall i, 1 \le i \le n: \map {\pi_i} x = \map {\pi_i} e} = \set {x \in G: \fo... | Let $G$ be a [[Definition:Group|group]].
Let $N_1, \ldots, N_n$ for some $n \ge 1$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$.
Let $\pi_i: G \rightarrow G / N_i$ be the [[Definition:Quotient Mapping|canonical projections]].
Then the [[Definition:Group Homomorphism|homomorphism]] $\pi: G \to G / N_1 \t... | By [[Quotient Group Epimorphism is Epimorphism]], each [[Definition:Quotient Epimorphism|canonical projection]] $\pi_i: G \to G / N_i$ is a [[Definition:Group Homomorphism|group homomorphism]].
Hence the mapping $\pi$ is indeed a [[Definition:Group Homomorphism|group homomorphism]].
The [[Definition:Kernel of Group H... | Chinese Remainder Theorem (Groups) | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Groups) | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Groups) | [
"Chinese Remainder Theorem",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Mapping",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Surjection",
"Definition:Normal Subgroup"
] | [
"Quotient Epimorphism is Epimorphism/Group",
"Definition:Quotient Epimorphism",
"Definition:Group Homomorphism",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definiti... |
proofwiki-19907 | Square Modulo 24 of Odd Integer Not Divisible by 3 | Let $a \in \Z$ be an integer such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-divisibility.
Then:
:$a^2 \equiv 1 \pmod {24}$
That is:
:$24 \divides \paren {a^2 - 1}$
where $\divides$ denotes divisibility. | Let $a$ be as asserted.
We have that:
:$2 \nmid a$
From Odd Square Modulo 8:
:$a^2 \equiv 1 \pmod 8$
which means:
:$8 \divides a^2 - 1$
We also have that:
:$3 \nmid a$
From Square Modulo 3: Corollary 3:
:$3 \divides a^2 - 1$
We have from Coprime Integers: $3$ and $8$ that:
:$3 \perp 8$
where $\perp$ denotes coprimality... | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
Then:
:$a^2 \equiv 1 \pmod {24}$
That is:
:$24 \divides \paren {a^2 - 1}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | Let $a$ be as asserted.
We have that:
:$2 \nmid a$
From [[Odd Square Modulo 8]]:
:$a^2 \equiv 1 \pmod 8$
which means:
:$8 \divides a^2 - 1$
We also have that:
:$3 \nmid a$
From [[Square Modulo 3/Corollary 3|Square Modulo 3: Corollary 3]]:
:$3 \divides a^2 - 1$
We have from [[Coprime Integers/Examples/3 and 8|... | Square Modulo 24 of Odd Integer Not Divisible by 3/Proof 1 | https://proofwiki.org/wiki/Square_Modulo_24_of_Odd_Integer_Not_Divisible_by_3 | https://proofwiki.org/wiki/Square_Modulo_24_of_Odd_Integer_Not_Divisible_by_3/Proof_1 | [
"Square Modulo 24 of Odd Integer Not Divisible by 3",
"Odd Squares"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Odd Square Modulo 8",
"Square Modulo 3/Corollary 3",
"Coprime Integers/Examples/3 and 8",
"Definition:Coprime/Integers",
"Product of Coprime Factors"
] |
proofwiki-19908 | Sum of Squares of Two Odd Integers is not Square | Let $m$ and $n$ be odd integers.
Then $m^2 + n^2$ is not a square number. | {{AimForCont}} $m^2 + n^2$ is a square number.
Because $m$ and $n$ are both odd, we have:
{{begin-eqn}}
{{eqn | l = m^2
| o = \equiv
| r = 1
| rr= \pmod 4
| c = Square Modulo 4
}}
{{eqn | l = n^2
| o = \equiv
| r = 1
| rr= \pmod 4
| c = Square Modulo 4
}}
{{eqn | ll= \le... | Let $m$ and $n$ be [[Definition:Odd Integer|odd integers]].
Then $m^2 + n^2$ is not a [[Definition:Square Number|square number]]. | {{AimForCont}} $m^2 + n^2$ is a [[Definition:Square Number|square number]].
Because $m$ and $n$ are both [[Definition:Odd Integer|odd]], we have:
{{begin-eqn}}
{{eqn | l = m^2
| o = \equiv
| r = 1
| rr= \pmod 4
| c = [[Square Modulo 4]]
}}
{{eqn | l = n^2
| o = \equiv
| r = 1
... | Sum of Squares of Two Odd Integers is not Square | https://proofwiki.org/wiki/Sum_of_Squares_of_Two_Odd_Integers_is_not_Square | https://proofwiki.org/wiki/Sum_of_Squares_of_Two_Odd_Integers_is_not_Square | [
"Odd Squares",
"Sums of Squares"
] | [
"Definition:Odd Integer",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Odd Integer",
"Square Modulo 4",
"Square Modulo 4",
"Parity of Integer equals Parity of its Square",
"Definition:Odd Integer",
"Definition:Even Integer",
"Square Modulo 4",
"Definition:Even Integer",
"Definition:Contradiction",
"Proof by Contradiction",
... |
proofwiki-19909 | Polygon Triangulation Theorem | Let $P$ be a polygon with $n$ sides, where $n \in \N_{ \ge 3 }$.
Then there exists a triangulation of $P$ that fulfills this condition:
* If $AB$ is a side of a triangle in the triangulation of $P$, then $AB$ is either a side of $P$, or a chord of $P$ that lies completely in the interior of $P$.
All triangulations of $... | We show existence by using strong induction over $n$, the number of sides of $P$. | Let $P$ be a [[Definition:Polygon|polygon]] with $n$ [[Definition:Side of Polygon|sides]], where $n \in \N_{ \ge 3 }$.
Then there exists a [[Definition:Triangulation of Polygon|triangulation]] of $P$ that fulfills this condition:
* If $AB$ is a [[Definition:Side of Polygon|side]] of a [[Definition:Triangle (Geometry... | We show existence by using [[Principle of Strong Induction|strong induction]] over $n$, the number of [[Definition:Side of Polygon|sides]] of $P$. | Polygon Triangulation Theorem | https://proofwiki.org/wiki/Polygon_Triangulation_Theorem | https://proofwiki.org/wiki/Polygon_Triangulation_Theorem | [
"Triangles",
"Named Theorems"
] | [
"Definition:Polygon",
"Definition:Polygon/Side",
"Definition:Triangulation of Polygon",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Triangulation of Polygon",
"Definition:Polygon/Side",
"Definition:Polygon/Chord",
"Definition:Region",
"Definition:Triangulation of Polyg... | [
"Second Principle of Mathematical Induction",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Defini... |
proofwiki-19910 | Product of Consecutive Integers is Even | Let $a$ and $b$ be consecutive integers.
Then $a b$ is even. | {{WLOG}} let $a < b$.
Then $b$ can be expressed as $a + 1$.
Hence:
{{begin-eqn}}
{{eqn | l = a b
| r = a \paren {a + 1}
| c =
}}
{{eqn | r = a^2 + a
| c =
}}
{{end-eqn}}
From Parity of Integer equals Parity of its Square, $a$ and $a^2$ are either both even or both odd.
The result follows from:
:Sum ... | Let $a$ and $b$ be consecutive [[Definition:Integer|integers]].
Then $a b$ is [[Definition:Even Integer|even]]. | {{WLOG}} let $a < b$.
Then $b$ can be expressed as $a + 1$.
Hence:
{{begin-eqn}}
{{eqn | l = a b
| r = a \paren {a + 1}
| c =
}}
{{eqn | r = a^2 + a
| c =
}}
{{end-eqn}}
From [[Parity of Integer equals Parity of its Square]], $a$ and $a^2$ are either both [[Definition:Even Integer|even]] or both ... | Product of Consecutive Integers is Even | https://proofwiki.org/wiki/Product_of_Consecutive_Integers_is_Even | https://proofwiki.org/wiki/Product_of_Consecutive_Integers_is_Even | [
"Even Integers"
] | [
"Definition:Integer",
"Definition:Even Integer"
] | [
"Parity of Integer equals Parity of its Square",
"Definition:Even Integer",
"Definition:Odd Integer",
"Sum of Even Integers is Even",
"Sum of Even Number of Odd Numbers is Even",
"Category:Even Integers"
] |
proofwiki-19911 | GCD with Self | Let $a \in \Z$ be an integer such that $a \ne 0$.
Then:
:$\gcd \set {a, a} = \size a$
where $\gcd$ denotes greatest common divisor (GCD). | From Integer Divides its Absolute Value:
:$\size a \divides a$
Then from Absolute Value of Integer is not less than Divisors:
:$\forall x \in \Z: x \divides a \implies x \le \size a$
The result follows by definition of GCD.
{{qed}} | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that $a \ne 0$.
Then:
:$\gcd \set {a, a} = \size a$
where $\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor (GCD)]]. | From [[Integer Divides its Absolute Value]]:
:$\size a \divides a$
Then from [[Absolute Value of Integer is not less than Divisors]]:
:$\forall x \in \Z: x \divides a \implies x \le \size a$
The result follows by definition of [[Definition:Greatest Common Divisor of Integers|GCD]].
{{qed}} | GCD with Self | https://proofwiki.org/wiki/GCD_with_Self | https://proofwiki.org/wiki/GCD_with_Self | [
"Greatest Common Divisor"
] | [
"Definition:Integer",
"Definition:Greatest Common Divisor/Integers"
] | [
"Integer Divisor Results/Integer Divides its Absolute Value",
"Absolute Value of Integer is not less than Divisors",
"Definition:Greatest Common Divisor/Integers"
] |
proofwiki-19912 | GCD of Integer with Integer + n | Let $a \in \Z$ be an integer.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\gcd \set {a, a + n} \divides n$
where:
:$\gcd$ denotes the greatest common divisor
:$\divides$ denotes divisibility. | Let $g = \gcd \set {a, a + n}$.
By definition of $\gcd$, there exist $b, b' \in \Z$ such that:
:$a = g b$
:$a + n = g b'$
Therefore:
{{begin-eqn}}
{{eqn | l = n
| r = \paren{ a + n } - a
}}
{{eqn | r = g b' - g b
}}
{{eqn | r = g \paren{ b' - b }
}}
{{end-eqn}}
Since $b' - b \in \Z$, it follows by definition of d... | Let $a \in \Z$ be an [[Definition:Integer|integer]].
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\gcd \set {a, a + n} \divides n$
where:
:$\gcd$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]
:$\divides$ denotes [[Definition:Divisor of... | Let $g = \gcd \set {a, a + n}$.
By definition of $\gcd$, there exist $b, b' \in \Z$ such that:
:$a = g b$
:$a + n = g b'$
Therefore:
{{begin-eqn}}
{{eqn | l = n
| r = \paren{ a + n } - a
}}
{{eqn | r = g b' - g b
}}
{{eqn | r = g \paren{ b' - b }
}}
{{end-eqn}}
Since $b' - b \in \Z$, it follows by definition... | GCD of Integer with Integer + n | https://proofwiki.org/wiki/GCD_of_Integer_with_Integer_+_n | https://proofwiki.org/wiki/GCD_of_Integer_with_Integer_+_n | [
"Greatest Common Divisor"
] | [
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Greatest Common Divisor/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-19913 | Two Ears Theorem | Let $P$ be a polygon that is not a triangle.
Then $P$ has at least two ears that do not overlap each other. | Let $P$ have $n$ sides, where $n \in \N_{ \ge 3 }$.
Polygon Triangulation Theorem shows that there exists a triangulation $\family { \triangle_i }_{i \mathop = 1}^{n-2}$ of $P$, and the sides of each triangle $\triangle_i$ are either sides of $P$ or chords of $P$.
Suppose $\triangle_i$ has $0$ or $1$ sides in common wi... | Let $P$ be a [[Definition:Polygon|polygon]] that is not a [[Definition:Triangle (Geometry)|triangle]].
Then $P$ has at least two [[Definition:Ear|ears]] that do not overlap each other. | Let $P$ have $n$ [[Definition:Side of Polygon|sides]], where $n \in \N_{ \ge 3 }$.
[[Polygon Triangulation Theorem]] shows that there exists a [[Definition:Triangulation of Polygon|triangulation]] $\family { \triangle_i }_{i \mathop = 1}^{n-2}$ of $P$, and the [[Definition:Side of Polygon|sides]] of each [[Definition:... | Two Ears Theorem | https://proofwiki.org/wiki/Two_Ears_Theorem | https://proofwiki.org/wiki/Two_Ears_Theorem | [
"Triangles",
"Polygons"
] | [
"Definition:Polygon",
"Definition:Triangle (Geometry)",
"Definition:Ear"
] | [
"Definition:Polygon/Side",
"Polygon Triangulation Theorem",
"Definition:Triangulation of Polygon",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Polygon/Chord",
"Definition:Polygon/Side",
"Definition:Ear",
"Definition:Polygon/Side",
"Definiti... |
proofwiki-19914 | Intersection of Ideals of Ring contains Product | Let $R$ be a ring.
Let $I$ be a right ideal of $R$.
Let $J$ be a left ideal of $R$.
Let $I J$ be their product.
Then $I J \subseteq I \cap J$. | Let $a_1, \ldots, a_n \in I$ and $b_1, \ldots, b_n \in J$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | q = \forall k \in \set {1, \ldots, n}
| l = a_k b_k
| o = \in
| r = I
| c = {{Defof|Right Ideal of Ring}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^n a_k b_k
| o = \in
|... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $I$ be a [[Definition:Right Ideal of Ring|right ideal]] of $R$.
Let $J$ be a [[Definition:Left Ideal of Ring|left ideal]] of $R$.
Let $I J$ be their [[Definition:Product of Ideals of Ring|product]].
Then $I J \subseteq I \cap J$. | Let $a_1, \ldots, a_n \in I$ and $b_1, \ldots, b_n \in J$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | q = \forall k \in \set {1, \ldots, n}
| l = a_k b_k
| o = \in
| r = I
| c = {{Defof|Right Ideal of Ring}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^n a_k b_k
| o = \in
... | Intersection of Ideals of Ring contains Product | https://proofwiki.org/wiki/Intersection_of_Ideals_of_Ring_contains_Product | https://proofwiki.org/wiki/Intersection_of_Ideals_of_Ring_contains_Product | [
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring/Right Ideal",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Product of Ideals of Ring"
] | [
"Definition:Set Intersection",
"Category:Ideal Theory"
] |
proofwiki-19915 | Universal Instantiation/Proof System | Let $\map {\mathbf A} x$ be a WFF of predicate logic.
Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.
Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.
Then:
:$\forall x: \map {\mathbf A} x \vdash_{\mathscr H} \map {\mathbf A} \tau$
is a provable consequence in $\mathsc... | {{WIP}} | Let $\map {\mathbf A} x$ be a [[Definition:WFF of Predicate Logic|WFF of predicate logic]].
Let $\tau$ be a [[Definition:Term (Predicate Logic)|term]] which is [[Definition:Freely Substitutable|freely substitutable]] for $x$ in $\mathbf A$.
Let $\mathscr H$ be [[Definition:Hilbert Proof System Instance 1 for Predicat... | {{WIP}} | Universal Instantiation/Proof System | https://proofwiki.org/wiki/Universal_Instantiation/Proof_System | https://proofwiki.org/wiki/Universal_Instantiation/Proof_System | [
"Universal Instantiation",
"Hilbert Proof System Instance 1 for Predicate Logic"
] | [
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Freely Substitutable",
"Definition:Hilbert Proof System/Predicate Logic/Instance 1",
"Definition:Provable Consequence"
] | [] |
proofwiki-19916 | Universal Generalisation/Informal Statement | Let $\mathbf a$ be any arbitrarily selected object in the universe of discourse.
Then:
{{begin-eqn}}
{{eqn | l = \map P {\mathbf a}
| o =
}}
{{eqn | ll= \vdash
| q = \forall x
| l = \map P x
| o =
}}
{{end-eqn}}
In natural language:
:''Suppose $P$ is true of any arbitrarily selected $\mathbf a$'... | We can express $\forall x$ using its propositional expansion:
:$\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$
where $\mathbf X_1, \mathbf X_2, \mathbf X_3 \ldots{}$ is the complete set of the objects in the universe of discourse.
The fact that any object we care to choose has ... | Let $\mathbf a$ be any [[Definition:Arbitrary|arbitrarily]] selected [[Definition:Object|object]] in the [[Definition:Universe of Discourse|universe of discourse]].
Then:
{{begin-eqn}}
{{eqn | l = \map P {\mathbf a}
| o =
}}
{{eqn | ll= \vdash
| q = \forall x
| l = \map P x
| o =
}}
{{end-eqn}}... | We can express $\forall x$ using its [[Definition:Propositional Expansion|propositional expansion]]:
:$\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$
where $\mathbf X_1, \mathbf X_2, \mathbf X_3 \ldots{}$ is the complete [[Definition:Set|set]] of the [[Definition:Object|objects... | Universal Generalisation/Informal Statement | https://proofwiki.org/wiki/Universal_Generalisation/Informal_Statement | https://proofwiki.org/wiki/Universal_Generalisation/Informal_Statement | [
"Predicate Logic"
] | [
"Definition:Arbitrary",
"Definition:Object",
"Definition:Universe of Discourse",
"Definition:Natural Language",
"Definition:True",
"Definition:Arbitrary",
"Definition:Universe of Discourse",
"Definition:True",
"Definition:Universe of Discourse"
] | [
"Definition:Propositional Expansion",
"Definition:Set",
"Definition:Object",
"Definition:Universe of Discourse",
"Definition:Object",
"Definition:Property",
"Definition:Property",
"Rule of Conjunction",
"Definition:Universe of Discourse",
"Definition:Propositional Expansion"
] |
proofwiki-19917 | Universal Generalisation/Proof System | Let $\LL$ be a specific signature for the language of predicate logic.
Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.
Let $\map {\mathbf A} x$ be a WFF of $\LL$.
Let $\FF$ be a collection of WFFs of $\LL$.
Let $c$ be an arbitrary constant symbol which is not in $\LL$.
Let $\LL'$ be the signatu... | {{WIP}} | Let $\LL$ be a specific [[Definition:Signature for Predicate Logic|signature]] for the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $\mathscr H$ be [[Definition:Hilbert Proof System Instance 1 for Predicate Logic|Hilbert proof system instance 1 for predicate logic]].
Let $\map {\mathbf... | {{WIP}} | Universal Generalisation/Proof System | https://proofwiki.org/wiki/Universal_Generalisation/Proof_System | https://proofwiki.org/wiki/Universal_Generalisation/Proof_System | [
"Predicate Logic",
"Hilbert Proof System Instance 1 for Predicate Logic"
] | [
"Definition:Signature (Logic)/Predicate Logic",
"Definition:Language of Predicate Logic",
"Definition:Hilbert Proof System/Predicate Logic/Instance 1",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Constant Symbol",
"Definit... | [] |
proofwiki-19918 | Existential Instantiation/Informal Statement | :$\exists x: \map P x, \map P {\mathbf a} \implies y \vdash y$
Suppose we have the following:
:From our universe of discourse, ''any'' arbitrarily selected object $\mathbf a$ which has the property $P$ implies a conclusion $y$
:$\mathbf a$ is not free in $y$
:It is known that there ''does'' actually exists an object th... | This is an extension of Proof by Cases.
The propositional expansion of $\exists x: \map P x$ is:
:$\map P {\mathbf X_1} \lor \map P {\mathbf X_2} \lor \map P {\mathbf X_3} \lor \ldots$
We know that any arbitrarily selected $\mathbf a$ with the property $P$ implies $y$.
From this we can infer that ''all'' such $\mathbf ... | :$\exists x: \map P x, \map P {\mathbf a} \implies y \vdash y$
Suppose we have the following:
:From our [[Definition:Universe of Discourse|universe of discourse]], ''any'' arbitrarily selected [[Definition:Object|object]] $\mathbf a$ which has the [[Definition:Property|property]] $P$ implies a conclusion $y$
:$\mathb... | This is an extension of [[Proof by Cases]].
The [[Definition:Propositional Expansion|propositional expansion]] of $\exists x: \map P x$ is:
:$\map P {\mathbf X_1} \lor \map P {\mathbf X_2} \lor \map P {\mathbf X_3} \lor \ldots$
We know that any arbitrarily selected $\mathbf a$ with the property $P$ implies $y$.
Fro... | Existential Instantiation/Informal Statement | https://proofwiki.org/wiki/Existential_Instantiation/Informal_Statement | https://proofwiki.org/wiki/Existential_Instantiation/Informal_Statement | [
"Predicate Logic"
] | [
"Definition:Universe of Discourse",
"Definition:Object",
"Definition:Property",
"Definition:Free Variable"
] | [
"Proof by Cases",
"Definition:Propositional Expansion",
"Proof by Cases",
"Definition:Disjunction/Disjunct",
"Definition:Propositional Expansion"
] |
proofwiki-19919 | Existential Instantiation/Proof System | Let $\LL$ be a specific signature for the language of predicate logic.
Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.
Let $\map {\mathbf A} x, \mathbf B$ be WFFs of $\LL$.
Let $\FF$ be a collection of WFFs of $\LL$.
Let $c$ be an arbitrary constant symbol which is not in $\LL$.
Let $\LL'$ be t... | {{WIP}} | Let $\LL$ be a specific [[Definition:Signature for Predicate Logic|signature]] for the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $\mathscr H$ be [[Definition:Hilbert Proof System Instance 1 for Predicate Logic|Hilbert proof system instance 1 for predicate logic]].
Let $\map {\mathbf... | {{WIP}} | Existential Instantiation/Proof System | https://proofwiki.org/wiki/Existential_Instantiation/Proof_System | https://proofwiki.org/wiki/Existential_Instantiation/Proof_System | [
"Predicate Logic",
"Hilbert Proof System Instance 1 for Predicate Logic"
] | [
"Definition:Signature (Logic)/Predicate Logic",
"Definition:Language of Predicate Logic",
"Definition:Hilbert Proof System/Predicate Logic/Instance 1",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Constant Symbol",
"Definit... | [] |
proofwiki-19920 | Components of Separation are Separated Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \mid B$ be a separation of $T$.
Then $A$ and $B$ are separated sets of $T$. | By definition of closure, $A^-$ is the smallest closed set of $T$ that contains $A$.
Components of Separation are Clopen shows that $A$ and $B$ are closed.
It follows that $A^- = A$, and $B^- = B$.
Definition of separation shows that $A \cap B = \O$, so we have:
{{begin-eqn}}
{{eqn | l = A^- \cap B
| r = A \cap B... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \mid B$ be a [[Definition:Separation (Topology)|separation]] of $T$.
Then $A$ and $B$ are [[Definition:Separated Sets|separated sets]] of $T$. | By definition of [[Definition:Closure (Topology)/Definition 3|closure]], $A^-$ is the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Closed Set (Topology)|closed set]] of $T$ that [[Definition:Subset|contains]] $A$.
[[Components of Separation are Clopen]] shows that $A$ and $B$ are [[Definition:Clo... | Components of Separation are Separated Sets | https://proofwiki.org/wiki/Components_of_Separation_are_Separated_Sets | https://proofwiki.org/wiki/Components_of_Separation_are_Separated_Sets | [
"Separated Sets",
"Separations"
] | [
"Definition:Topological Space",
"Definition:Separation (Topology)",
"Definition:Separated Sets"
] | [
"Definition:Closure (Topology)/Definition 3",
"Definition:Smallest Set by Set Inclusion",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Components of Separation are Clopen",
"Definition:Closed Set/Topology",
"Definition:Separation (Topology)",
"Definition:Separated Sets",
"Category:Separat... |
proofwiki-19921 | Integer Combination of Coprime Integers/Sufficient Condition | Let $a, b \in \Z$ be integers, not both zero.
Let $a$ and $b$ be coprime to each other.
Then there exists an integer combination of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exists m, n \in \Z: m a + n b = 1$ | {{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| r = 1
| c = {{Defof|Coprime Integers}}
}}
{{eqn | ll= \leadsto
| q = \exists m, n \in \Z
| l = m a + n b
| r = 1
| c = Bézout's Identity
}}
{{end-eqn}}
{{qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]], not both [[Definition:Zero (Number)|zero]].
Let $a$ and $b$ be [[Definition:Coprime Integers|coprime]] to each other.
Then there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exist... | {{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| r = 1
| c = {{Defof|Coprime Integers}}
}}
{{eqn | ll= \leadsto
| q = \exists m, n \in \Z
| l = m a + n b
| r = 1
| c = [[Bézout's Identity]]
}}
{{end-eqn}}
{{qed}} | Integer Combination of Coprime Integers/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition/Proof_1 | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Coprime/Integers",
"Definition:Integer Combination"
] | [
"Bézout's Identity"
] |
proofwiki-19922 | Integer Combination of Coprime Integers/Sufficient Condition | Let $a, b \in \Z$ be integers, not both zero.
Let $a$ and $b$ be coprime to each other.
Then there exists an integer combination of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exists m, n \in \Z: m a + n b = 1$ | Let $a \perp b$.
Thus they are not both $0$.
Let $S$ be defined as:
:$S = \set {a m + b n: m, n \in \Z}$
$S$ contains at least one strictly positive integer, because for example $a^2 + b^2 \in S$.
By Set of Integers Bounded Below has Smallest Element, let $d$ be the smallest element of $S$ which is strictly positive.
L... | Let $a, b \in \Z$ be [[Definition:Integer|integers]], not both [[Definition:Zero (Number)|zero]].
Let $a$ and $b$ be [[Definition:Coprime Integers|coprime]] to each other.
Then there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exist... | Let $a \perp b$.
Thus they are not both $0$.
Let $S$ be defined as:
:$S = \set {a m + b n: m, n \in \Z}$
$S$ contains at least one [[Definition:Strictly Positive Integer|strictly positive integer]], because for example $a^2 + b^2 \in S$.
By [[Set of Integers Bounded Below has Smallest Element]], let $d$ be the [[De... | Integer Combination of Coprime Integers/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition/Proof_2 | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Coprime/Integers",
"Definition:Integer Combination"
] | [
"Definition:Strictly Positive/Integer",
"Set of Integers Bounded Below has Smallest Element",
"Definition:Smallest Element",
"Definition:Strictly Positive/Integer",
"Division Theorem",
"Definition:Smallest Element",
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Integer",
"Def... |
proofwiki-19923 | Integer Combination of Coprime Integers/Sufficient Condition | Let $a, b \in \Z$ be integers, not both zero.
Let $a$ and $b$ be coprime to each other.
Then there exists an integer combination of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exists m, n \in \Z: m a + n b = 1$ | Let $a \perp b$.
Thus they are not both $0$.
Let $S$ be defined as:
:$S = \set {\lambda a + \mu b: \lambda, \mu \in \Z}$
$S$ contains at least one strictly positive integer, because for example:
:$a \in S$ (setting $\lambda = 1$ and $\mu = 0$)
:$b \in S$ (setting $\lambda = 0$ and $\mu = 1$)
By Set of Integers Bounded ... | Let $a, b \in \Z$ be [[Definition:Integer|integers]], not both [[Definition:Zero (Number)|zero]].
Let $a$ and $b$ be [[Definition:Coprime Integers|coprime]] to each other.
Then there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$:
:$\forall a, b \in \Z: a \perp b \implies \exist... | Let $a \perp b$.
Thus they are not both $0$.
Let $S$ be defined as:
:$S = \set {\lambda a + \mu b: \lambda, \mu \in \Z}$
$S$ contains at least one [[Definition:Strictly Positive Integer|strictly positive integer]], because for example:
:$a \in S$ (setting $\lambda = 1$ and $\mu = 0$)
:$b \in S$ (setting $\lambda = 0... | Integer Combination of Coprime Integers/Sufficient Condition/Proof 3 | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Sufficient_Condition/Proof_3 | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Coprime/Integers",
"Definition:Integer Combination"
] | [
"Definition:Strictly Positive/Integer",
"Set of Integers Bounded Below has Smallest Element",
"Definition:Smallest Element",
"Definition:Strictly Positive/Integer",
"Division Theorem",
"Definition:Smallest Element",
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Integer",
"Def... |
proofwiki-19924 | Integer Combination of Coprime Integers/Necessary Condition | Let $a, b \in \Z$ be integers, not both zero.
Let $a$ and $b$ be such that there exists an integer combination of them equal to $1$.
Then $a$ and $b$ are coprime:
:$\forall a, b \in \Z: \exists m, n \in \Z: m a + n b = 1 \implies a \perp b$
In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are al... | {{begin-eqn}}
{{eqn | q = \exists m, n \in \Z
| l = m a + n b
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| o = \divides
| r = 1
| c = Set of Integer Combinations equals Set of Multiples of GCD
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| r = 1... | Let $a, b \in \Z$ be [[Definition:Integer|integers]], not both [[Definition:Zero (Number)|zero]].
Let $a$ and $b$ be such that there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$.
Then $a$ and $b$ are [[Definition:Coprime Integers|coprime]]:
:$\forall a, b \in \Z: \exists m, n... | {{begin-eqn}}
{{eqn | q = \exists m, n \in \Z
| l = m a + n b
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| o = \divides
| r = 1
| c = [[Set of Integer Combinations equals Set of Multiples of GCD]]
}}
{{eqn | ll= \leadsto
| l = \gcd \set {a, b}
| r... | Integer Combination of Coprime Integers/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Necessary_Condition | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Necessary_Condition/Proof_1 | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Integer Combination",
"Definition:Coprime/Integers",
"Definition:Integer Combination",
"Definition:Integer",
"Definition:Coprime/Integers"
] | [
"Set of Integer Combinations equals Set of Multiples of GCD"
] |
proofwiki-19925 | Integer Combination of Coprime Integers/Necessary Condition | Let $a, b \in \Z$ be integers, not both zero.
Let $a$ and $b$ be such that there exists an integer combination of them equal to $1$.
Then $a$ and $b$ are coprime:
:$\forall a, b \in \Z: \exists m, n \in \Z: m a + n b = 1 \implies a \perp b$
In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are al... | Let $a, b \in \Z$ be such that $\exists m, n \in \Z: m a + n b = 1$.
Let $d$ be a divisor of both $a$ and $b$.
Then:
:$d \divides m a + n b$
and so:
:$d \divides 1$
Thus:
:$d = \pm 1$
and so:
:$\gcd \set {a, b} = 1$
Thus, by definition, $a$ and $b$ are coprime.
{{qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]], not both [[Definition:Zero (Number)|zero]].
Let $a$ and $b$ be such that there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$.
Then $a$ and $b$ are [[Definition:Coprime Integers|coprime]]:
:$\forall a, b \in \Z: \exists m, n... | Let $a, b \in \Z$ be such that $\exists m, n \in \Z: m a + n b = 1$.
Let $d$ be a [[Definition:Divisor of Integer|divisor]] of both $a$ and $b$.
Then:
:$d \divides m a + n b$
and so:
:$d \divides 1$
Thus:
:$d = \pm 1$
and so:
:$\gcd \set {a, b} = 1$
Thus, by definition, $a$ and $b$ are [[Definition:Coprime Intege... | Integer Combination of Coprime Integers/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Necessary_Condition | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/Necessary_Condition/Proof_2 | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Integer Combination",
"Definition:Coprime/Integers",
"Definition:Integer Combination",
"Definition:Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers"
] |
proofwiki-19926 | Connected Subspace lies in One Component of Separation | Let $T = \struct {S, \tau}$ be a topological space.
Let $A_1 \mid A_2$ be a separation of $T$.
Let $X$ be a connected set of $T$.
Then there exist $i, j \in \set {1, 2}$ with $i \ne j$ such that $X \subseteq A_i$, and $X \cap A_j = \O$. | By definition of separation, $A_1$ and $A_2$ are disjoint.
Define $B_i = X \cap A_i$ for $i \in \set {1, 2}$.
From Intersection is Subset, $B_i \subseteq A_i$ for $i \in \set {1, 2}$.
From Subsets of Disjoint Sets are Disjoint, $B_1$ and $B_2$ are disjoint.
The union of $B_1$ and $B_2$ is:
{{begin-eqn}}
{{eqn | l = B_1... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A_1 \mid A_2$ be a [[Definition:Separation (Topology)|separation]] of $T$.
Let $X$ be a [[Definition:Connected Set (Topology)|connected set]] of $T$.
Then there exist $i, j \in \set {1, 2}$ with $i \ne j$ such that $X \subsete... | By definition of [[Definition:Separation (Topology)|separation]], $A_1$ and $A_2$ are [[Definition:Disjoint Sets|disjoint]].
Define $B_i = X \cap A_i$ for $i \in \set {1, 2}$.
From [[Intersection is Subset]], $B_i \subseteq A_i$ for $i \in \set {1, 2}$.
From [[Subsets of Disjoint Sets are Disjoint]], $B_1$ and $B_2$... | Connected Subspace lies in One Component of Separation | https://proofwiki.org/wiki/Connected_Subspace_lies_in_One_Component_of_Separation | https://proofwiki.org/wiki/Connected_Subspace_lies_in_One_Component_of_Separation | [
"Connected Sets (Topology)",
"Separations"
] | [
"Definition:Topological Space",
"Definition:Separation (Topology)",
"Definition:Connected Set (Topology)"
] | [
"Definition:Separation (Topology)",
"Definition:Disjoint Sets",
"Intersection is Subset",
"Subsets of Disjoint Sets are Disjoint",
"Definition:Disjoint Sets",
"Definition:Set Union",
"Intersection Distributes over Union",
"Intersection with Subset is Subset",
"Definition:Separation (Topology)",
"D... |
proofwiki-19927 | Supremum Operator Norm Need not be Attained | Let $\Bbb K \in \set {\R, \C}$.
Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a bounded sequence in $\Bbb K$ such that:
:$\ds \forall n \in \N_{> 0} : \lambda_n = 1 - \frac 1 n$.
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the normed $2$-sequence space.
Let $\mathbf x = \tuple {a_1, a_2, a_3, \ldots} \in \ell^... | Suppose $\ds \lambda_n = 1 - \frac 1 n$ with $n \in \N_{> 0}$.
By Supremum Operator Norm of Diagonal Operator over 2-Sequence Space:
{{begin-eqn}}
{{eqn | l = \norm \Lambda
| r = \sup_{n \mathop \in \N_{>0} } \set {1 - \frac 1 n}
}}
{{eqn | r = 1
}}
{{end-eqn}}
Suppose:
:$\mathbf x = \tuple {a_n}_{n \mathop \in... | Let $\Bbb K \in \set {\R, \C}$.
Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a [[Definition:Bounded Sequence|bounded sequence]] in $\Bbb K$ such that:
:$\ds \forall n \in \N_{> 0} : \lambda_n = 1 - \frac 1 n$.
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vecto... | Suppose $\ds \lambda_n = 1 - \frac 1 n$ with $n \in \N_{> 0}$.
By [[Supremum Operator Norm of Diagonal Operator over 2-Sequence Space]]:
{{begin-eqn}}
{{eqn | l = \norm \Lambda
| r = \sup_{n \mathop \in \N_{>0} } \set {1 - \frac 1 n}
}}
{{eqn | r = 1
}}
{{end-eqn}}
Suppose:
:$\mathbf x = \tuple {a_n}_{n \ma... | Supremum Operator Norm Need not be Attained | https://proofwiki.org/wiki/Supremum_Operator_Norm_Need_not_be_Attained | https://proofwiki.org/wiki/Supremum_Operator_Norm_Need_not_be_Attained | [
"Supremum Norm",
"Continuous Linear Transformations"
] | [
"Definition:Bounded Sequence",
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Diagonal Operator"
] | [
"Supremum Operator Norm of Diagonal Operator over 2-Sequence Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:Term of Sequence",
"Definition:Contradiction"
] |
proofwiki-19928 | Coefficients in Linear Combination forming GCD are Coprime | Let $a$ and $b$ be integers.
Let there exist integers $x$ and $y$ such that:
:$a x + b y = \gcd \set {a, b}$
where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
Then:
:$x \perp y$
where $\perp$ denotes coprimality. | Let $d = \gcd \set {a, b}$.
As $d$ is a divisor of both $a$ and $b$, both $\dfrac a d$ and $\dfrac b d$ are integers.
Hence, dividing through by $d$, we have:
:$\dfrac a d x + \dfrac b d y = \dfrac {\gcd \set {a, b} } d = 1$
Thus there exist integers $m = \dfrac a d$ and $n = \dfrac b d$ such that:
:$m x + n y = 1$
Hen... | Let $a$ and $b$ be [[Definition:Integer|integers]].
Let there exist [[Definition:Integer|integers]] $x$ and $y$ such that:
:$a x + b y = \gcd \set {a, b}$
where $\gcd \set {a, b}$ denotes the [[Definition:Greatest Common Divisor|greatest common divisor]] of $a$ and $b$.
Then:
:$x \perp y$
where $\perp$ denotes [[Def... | Let $d = \gcd \set {a, b}$.
As $d$ is a [[Definition:Divisor of Integer|divisor]] of both $a$ and $b$, both $\dfrac a d$ and $\dfrac b d$ are [[Definition:Integer|integers]].
Hence, [[Definition:Integer Division|dividing]] through by $d$, we have:
:$\dfrac a d x + \dfrac b d y = \dfrac {\gcd \set {a, b} } d = 1$
Thu... | Coefficients in Linear Combination forming GCD are Coprime | https://proofwiki.org/wiki/Coefficients_in_Linear_Combination_forming_GCD_are_Coprime | https://proofwiki.org/wiki/Coefficients_in_Linear_Combination_forming_GCD_are_Coprime | [
"Integer Combinations",
"Greatest Common Divisor",
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Greatest Common Divisor",
"Definition:Coprime/Integers"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Integer Division",
"Definition:Integer",
"Integer Combination of Coprime Integers"
] |
proofwiki-19929 | Product of 3 Consecutive Integers is Divisible by 6 | Let $a, b, c \in \Z$ be consecutive integers.
Then their product $a b c$ is divisible by $6$. | This is an application of Divisibility of Product of Consecutive Integers with $n = 3$.
By the theorem, the product of $3$ consecutive integers is divisible by $3! = 6$.
{{qed}} | Let $a, b, c \in \Z$ be consecutive [[Definition:Integer|integers]].
Then their [[Definition:Integer Multiplication|product]] $a b c$ is [[Definition:Divisor of Integer|divisible]] by $6$. | This is an application of [[Divisibility of Product of Consecutive Integers]] with $n = 3$.
By the theorem, the [[Definition:Integer Multiplication|product]] of $3$ consecutive [[Definition:Integer|integers]] is [[Definition:Divisor of Integer|divisible]] by $3! = 6$.
{{qed}} | Product of 3 Consecutive Integers is Divisible by 6 | https://proofwiki.org/wiki/Product_of_3_Consecutive_Integers_is_Divisible_by_6 | https://proofwiki.org/wiki/Product_of_3_Consecutive_Integers_is_Divisible_by_6 | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisibility of Product of Consecutive Integers",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-19930 | Product of 4 Consecutive Integers is Divisible by 24 | Let $a, b, c, d \in \Z$ be consecutive integers.
Then their product $a b c d$ is divisible by $24$. | This is an application of Divisibility of Product of Consecutive Integers with $n = 4$.
By the theorem, the product of $4$ consecutive integers is divisible by $4! = 24$.
{{qed}} | Let $a, b, c, d \in \Z$ be consecutive [[Definition:Integer|integers]].
Then their [[Definition:Integer Multiplication|product]] $a b c d$ is [[Definition:Divisor of Integer|divisible]] by $24$. | This is an application of [[Divisibility of Product of Consecutive Integers]] with $n = 4$.
By the theorem, the [[Definition:Integer Multiplication|product]] of $4$ consecutive [[Definition:Integer|integers]] is [[Definition:Divisor of Integer|divisible]] by $4! = 24$.
{{qed}} | Product of 4 Consecutive Integers is Divisible by 24 | https://proofwiki.org/wiki/Product_of_4_Consecutive_Integers_is_Divisible_by_24 | https://proofwiki.org/wiki/Product_of_4_Consecutive_Integers_is_Divisible_by_24 | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisibility of Product of Consecutive Integers",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-19931 | Product of 5 Consecutive Integers is Divisible by 120 | Let $a, b, c, d, e \in Z$ be consecutive integers
Then their product $a b c d e$ is divisible by $120$. | This is an application of Divisibility of Product of Consecutive Integers with $n = 5$.
By the theorem, the product of $5$ consecutive integers is divisible by $5! = 120$.
{{qed}} | Let $a, b, c, d, e \in Z$ be consecutive [[Definition:Integer|integers]]
Then their [[Definition:Integer Multiplication|product]] $a b c d e$ is [[Definition:Divisor of Integer|divisible]] by $120$. | This is an application of [[Divisibility of Product of Consecutive Integers]] with $n = 5$.
By the theorem, the [[Definition:Integer Multiplication|product]] of $5$ consecutive [[Definition:Integer|integers]] is [[Definition:Divisor of Integer|divisible]] by $5! = 120$.
{{qed}} | Product of 5 Consecutive Integers is Divisible by 120 | https://proofwiki.org/wiki/Product_of_5_Consecutive_Integers_is_Divisible_by_120 | https://proofwiki.org/wiki/Product_of_5_Consecutive_Integers_is_Divisible_by_120 | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisibility of Product of Consecutive Integers",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-19932 | 24 divides a(a^2 - 1) when a is Odd | Let $a \in \Z$ be an odd integer.
Then:
:$24 \divides a \paren {a^2 - 1}$
where $\divides$ denotes divisibility. | First suppose that $a$ is not divisible by $3$.
Then from Square Modulo 24 of Odd Integer Not Divisible by 3:
:$24 \divides \paren {a^2 - 1}$
from which the result follows immediately.
{{qed|lemma}}
Now suppose that $3 \divides a$.
Then immediately:
:$3 \divides a \paren {a^2 - 1}$
From Odd Square Modulo 8:
:$8 \divide... | Let $a \in \Z$ be an [[Definition:Odd Integer|odd integer]].
Then:
:$24 \divides a \paren {a^2 - 1}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | First suppose that $a$ is not [[Definition:Divisor of Integer|divisible]] by $3$.
Then from [[Square Modulo 24 of Odd Integer Not Divisible by 3]]:
:$24 \divides \paren {a^2 - 1}$
from which the result follows immediately.
{{qed|lemma}}
Now suppose that $3 \divides a$.
Then immediately:
:$3 \divides a \paren {a^2 -... | 24 divides a(a^2 - 1) when a is Odd | https://proofwiki.org/wiki/24_divides_a(a^2_-_1)_when_a_is_Odd | https://proofwiki.org/wiki/24_divides_a(a^2_-_1)_when_a_is_Odd | [
"Examples of Divisors of Integers"
] | [
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Square Modulo 24 of Odd Integer Not Divisible by 3",
"Odd Square Modulo 8",
"Coprime Integers/Examples/3 and 8",
"Definition:Coprime/Integers",
"Product of Coprime Factors"
] |
proofwiki-19933 | 24 divides Square of Odd Integer Not Divisible by 3 plus 23 | Let $a \in \Z$ be an integer such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-divisibility.
Then:
:$24 \divides \paren {a^2 + 23}$
where $\divides$ denotes divisibility. | Let $a$ be as defined.
Then:
{{begin-eqn}}
{{eqn | q =
| l = 24
| o = \divides
| r = \paren {a^2 - 1}
| c = Square Modulo 24 of Odd Integer Not Divisible by 3
}}
{{eqn | ll= \leadsto
| l = 24
| o = \divides
| r = \paren {a^2 - 1 + 24}
| c =
}}
{{eqn | ll= \leadsto
... | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
Then:
:$24 \divides \paren {a^2 + 23}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | Let $a$ be as defined.
Then:
{{begin-eqn}}
{{eqn | q =
| l = 24
| o = \divides
| r = \paren {a^2 - 1}
| c = [[Square Modulo 24 of Odd Integer Not Divisible by 3]]
}}
{{eqn | ll= \leadsto
| l = 24
| o = \divides
| r = \paren {a^2 - 1 + 24}
| c =
}}
{{eqn | ll= \leadsto
... | 24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 1 | https://proofwiki.org/wiki/24_divides_Square_of_Odd_Integer_Not_Divisible_by_3_plus_23 | https://proofwiki.org/wiki/24_divides_Square_of_Odd_Integer_Not_Divisible_by_3_plus_23/Proof_1 | [
"24 divides Square of Odd Integer Not Divisible by 3 plus 23",
"Odd Squares"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Square Modulo 24 of Odd Integer Not Divisible by 3"
] |
proofwiki-19934 | 24 divides Square of Odd Integer Not Divisible by 3 plus 23 | Let $a \in \Z$ be an integer such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-divisibility.
Then:
:$24 \divides \paren {a^2 + 23}$
where $\divides$ denotes divisibility. | Let $a$ be as defined.
We have that $a$ is of the form:
:$a = 6 k + 1$
or:
:$a = 6 k + 5$
Hence:
{{begin-eqn}}
{{eqn | l = a^2 + 23
| r = \paren {6 k + 1}^2 + 23
| c =
}}
{{eqn | r = 36 k^2 + 12 k + 24
| c =
}}
{{eqn | r = 24 \paren {\dfrac {k \paren {3 k + 1} } 2 + 1}
| c =
}}
{{end-eqn}}
or... | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
Then:
:$24 \divides \paren {a^2 + 23}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | Let $a$ be as defined.
We have that $a$ is of the form:
:$a = 6 k + 1$
or:
:$a = 6 k + 5$
Hence:
{{begin-eqn}}
{{eqn | l = a^2 + 23
| r = \paren {6 k + 1}^2 + 23
| c =
}}
{{eqn | r = 36 k^2 + 12 k + 24
| c =
}}
{{eqn | r = 24 \paren {\dfrac {k \paren {3 k + 1} } 2 + 1}
| c =
}}
{{end-eqn}}... | 24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 2 | https://proofwiki.org/wiki/24_divides_Square_of_Odd_Integer_Not_Divisible_by_3_plus_23 | https://proofwiki.org/wiki/24_divides_Square_of_Odd_Integer_Not_Divisible_by_3_plus_23/Proof_2 | [
"24 divides Square of Odd Integer Not Divisible by 3 plus 23",
"Odd Squares"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-19935 | 360 divides a^2 (a^2 - 1) (a^2 - 4) | Let $a \in \Z$ be an integer.
Then:
:$360 \divides a^2 \paren {a^2 - 1} \paren {a^2 - 4}$
where $\divides$ denotes divisibility. | By Difference of Two Squares:
:$a^2 \paren {a^2 - 1} \paren {a^2 - 4} = a \paren {a - 2} \paren {a - 1} a \paren {a + 1} \paren {a + 2}$
We have that $a - 2, a - 1, a, a + 1, a + 2$ are $5$ consecutive integers.
Hence from Product of 5 Consecutive Integers is Divisible by 120:
:$120 \divides a \paren {a^2 - 1} \paren {... | Let $a \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$360 \divides a^2 \paren {a^2 - 1} \paren {a^2 - 4}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | By [[Difference of Two Squares]]:
:$a^2 \paren {a^2 - 1} \paren {a^2 - 4} = a \paren {a - 2} \paren {a - 1} a \paren {a + 1} \paren {a + 2}$
We have that $a - 2, a - 1, a, a + 1, a + 2$ are $5$ consecutive [[Definition:Integer|integers]].
Hence from [[Product of 5 Consecutive Integers is Divisible by 120]]:
:$120 \d... | 360 divides a^2 (a^2 - 1) (a^2 - 4) | https://proofwiki.org/wiki/360_divides_a^2_(a^2_-_1)_(a^2_-_4) | https://proofwiki.org/wiki/360_divides_a^2_(a^2_-_1)_(a^2_-_4) | [
"Examples of Divisors of Integers"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Difference of Two Squares",
"Definition:Integer",
"Product of 5 Consecutive Integers is Divisible by 120"
] |
proofwiki-19936 | GCD equals GCD with Product of Coprime Factor | Let $a, b, c \in \Z$ be integers.
Let:
:$a \perp b$
where $\perp$ denotes coprimality.
Then:
:$\gcd \set {a c, b} = \gcd \set {c, b}$
where $\gcd$ denotes greatest common divisor. | Let $a, b, c \in \Z$ such that $a$ is coprime to $b$.
Let $d = \gcd \set {c, b}$.
We have:
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| q = \exists x, y \in \Z
| l = 1
| r = a x + b y
| c = Integer Combination of Coprime Integer... | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let:
:$a \perp b$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then:
:$\gcd \set {a c, b} = \gcd \set {c, b}$
where $\gcd$ denotes [[Definition:Greatest Common Divisor|greatest common divisor]]. | Let $a, b, c \in \Z$ such that $a$ is [[Definition:Coprime Integers|coprime]] to $b$.
Let $d = \gcd \set {c, b}$.
We have:
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| q = \exists x, y \in \Z
| l = 1
| r = a x + b y
| c = [[I... | GCD equals GCD with Product of Coprime Factor | https://proofwiki.org/wiki/GCD_equals_GCD_with_Product_of_Coprime_Factor | https://proofwiki.org/wiki/GCD_equals_GCD_with_Product_of_Coprime_Factor | [
"Coprime Integers",
"Greatest Common Divisor"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Greatest Common Divisor"
] | [
"Definition:Coprime/Integers",
"Integer Combination of Coprime Integers",
"Bézout's Identity",
"Bézout's Identity"
] |
proofwiki-19937 | Existence of Chebyshev Polynomials of the First Kind | There exists a Chebyshev polynomial of the first kind for each natural number $n$. | For $n = 0$:
{{begin-eqn}}
{{eqn | l = \map \cos {0 \theta}
| r = \map \cos 0
}}
{{eqn | r = 1
}}
{{end-eqn}}
:$\map {T_0} x = 1$, $T_0 \in \Bbb P$
{{MissingLinks|$\Bbb P$ as the set/space of polynomials}}
For $n = 1$:
{{begin-eqn}}
{{eqn | l = \map \cos {1 \theta}
| r = \map \cos \theta
}}
{{end-eqn}}
:$\m... | There exists a [[Definition:Chebyshev Polynomial of the First Kind|Chebyshev polynomial of the first kind]] for each [[Definition:Natural Numbers|natural number]] $n$. | For $n = 0$:
{{begin-eqn}}
{{eqn | l = \map \cos {0 \theta}
| r = \map \cos 0
}}
{{eqn | r = 1
}}
{{end-eqn}}
:$\map {T_0} x = 1$, $T_0 \in \Bbb P$
{{MissingLinks|$\Bbb P$ as the set/space of polynomials}}
For $n = 1$:
{{begin-eqn}}
{{eqn | l = \map \cos {1 \theta}
| r = \map \cos \theta
}}
{{end-eqn}... | Existence of Chebyshev Polynomials of the First Kind | https://proofwiki.org/wiki/Existence_of_Chebyshev_Polynomials_of_the_First_Kind | https://proofwiki.org/wiki/Existence_of_Chebyshev_Polynomials_of_the_First_Kind | [
"Chebyshev Polynomials of the First Kind"
] | [
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Natural Numbers"
] | [
"Cosine of Sum",
"Cosine of Difference",
"Second Principle of Mathematical Induction",
"Category:Chebyshev Polynomials of the First Kind"
] |
proofwiki-19938 | Star Convex Set is Simply Connected | Let $A$ be a star convex subset of a topological vector space $\struct {V, \tau}$ over $\R$ or $\C$.
Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.
Then $\struct {A, \tau_A}$ is simply connected. | Let $a \in A$ be a star center of $A$.
Define $\mathbb I := \closedint 0 1$ as a closed real interval.
Let $\gamma : \mathbb I \to A$ be a loop in $A$ with base point $a$.
Let $\sigma : \mathbb I \to \set a$ be the constant function.
Constant Function is Continuous shows that $\sigma$ is continuous, so $\sigma$ is a lo... | Let $A$ be a [[Definition:Star Convex Set|star convex]] [[Definition:Subset|subset]] of a [[Definition:Topological Vector Space|topological vector space]] $\struct {V, \tau}$ over $\R$ or $\C$.
Let $\tau_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\tau$.
Then $\struct {A, \tau_A}$... | Let $a \in A$ be a [[Definition:Star Center|star center]] of $A$.
Define $\mathbb I := \closedint 0 1$ as a [[Definition:Closed Real Interval|closed real interval]].
Let $\gamma : \mathbb I \to A$ be a [[Definition:Loop (Topology)|loop]] in $A$ with [[Definition:Base Point of Loop|base point]] $a$.
Let $\sigma : \ma... | Star Convex Set is Simply Connected | https://proofwiki.org/wiki/Star_Convex_Set_is_Simply_Connected | https://proofwiki.org/wiki/Star_Convex_Set_is_Simply_Connected | [
"Vector Spaces",
"Simply Connected Spaces"
] | [
"Definition:Star Convex Set",
"Definition:Subset",
"Definition:Topological Vector Space",
"Definition:Topological Subspace",
"Definition:Simply Connected"
] | [
"Definition:Star Convex Set/Star Center",
"Definition:Real Interval/Closed",
"Definition:Loop (Topology)",
"Definition:Loop (Topology)/Base Point",
"Definition:Constant Mapping",
"Constant Function is Continuous",
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Loop (Topology)",
"Definit... |
proofwiki-19939 | Convex Set is Simply Connected | Let $\struct {V, \tau}$ be a topological vector space over $\R$ or $\C$.
Let $A \subseteq V$ be a non-empty convex set.
Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.
Then $\struct{ A, \tau_A }$ is simply connected. | Follows from Convex Set is Star Convex Set and Star Convex Set is Simply Connected.
{{qed}}
Category:Convex Sets (Vector Spaces)
Category:Simply Connected Spaces
ahzeja1h0tv4z4g66of67a273cw7i4d | Let $\struct {V, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$ or $\C$.
Let $A \subseteq V$ be a [[Definition:Empty Set|non-empty]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $\tau_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\t... | Follows from [[Convex Set is Star Convex Set]] and [[Star Convex Set is Simply Connected]].
{{qed}}
[[Category:Convex Sets (Vector Spaces)]]
[[Category:Simply Connected Spaces]]
ahzeja1h0tv4z4g66of67a273cw7i4d | Convex Set is Simply Connected | https://proofwiki.org/wiki/Convex_Set_is_Simply_Connected | https://proofwiki.org/wiki/Convex_Set_is_Simply_Connected | [
"Convex Sets (Vector Spaces)",
"Simply Connected Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Empty Set",
"Definition:Convex Set (Vector Space)",
"Definition:Topological Subspace",
"Definition:Simply Connected"
] | [
"Convex Set is Star Convex Set",
"Star Convex Set is Simply Connected",
"Category:Convex Sets (Vector Spaces)",
"Category:Simply Connected Spaces"
] |
proofwiki-19940 | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 1 | :$\neg \paren {\forall x: \map P x} \dashv \vdash \exists x: \neg \map P x$ | {{BeginTableau|\neg \paren {\forall x: \map P x} \vdash \exists x: \neg \map P x}}
{{Premise|1|\neg \paren {\forall x: \map P x} }}
{{Assumption|2|\neg \paren {\exists x: \neg \map P x} }}
{{Assumption|3|\neg \map P {\mathbf a}|for an arbitrary $\mathbf a$}}
{{TableauLine|n = 4|pool = 3|f = \exists x: \neg \map P x|rln... | :$\neg \paren {\forall x: \map P x} \dashv \vdash \exists x: \neg \map P x$ | {{BeginTableau|\neg \paren {\forall x: \map P x} \vdash \exists x: \neg \map P x}}
{{Premise|1|\neg \paren {\forall x: \map P x} }}
{{Assumption|2|\neg \paren {\exists x: \neg \map P x} }}
{{Assumption|3|\neg \map P {\mathbf a}|for an arbitrary $\mathbf a$}}
{{TableauLine|n = 4|pool = 3|f = \exists x: \neg \map P x|rln... | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 1 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_1 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_1 | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [] |
proofwiki-19941 | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2 | :$\vdash \neg \paren {\forall x: \map P x} \iff \paren{ \exists x: \neg \map P x }$ | {{WIP}} | :$\vdash \neg \paren {\forall x: \map P x} \iff \paren{ \exists x: \neg \map P x }$ | {{WIP}} | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2 | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [] |
proofwiki-19942 | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Forward Implication | :$\vdash \neg \paren {\forall x: \map P x} \implies \paren{ \exists x: \neg \map P x }$ | {{improve|These template invocations can be improved but I am happy the argument is at least conveyed now}}
{{BeginTableau|\vdash \neg \paren {\forall x: \map P x} \implies \paren{ \exists x: \neg \map P x }|Hilbert Proof System Instance 1 for Predicate Logic}}
{{Premise|1|\neg\neg \map P c \vdash \map P c | Axiom 1: P... | :$\vdash \neg \paren {\forall x: \map P x} \implies \paren{ \exists x: \neg \map P x }$ | {{improve|These template invocations can be improved but I am happy the argument is at least conveyed now}}
{{BeginTableau|\vdash \neg \paren {\forall x: \map P x} \implies \paren{ \exists x: \neg \map P x }|[[Definition:Hilbert Proof System Instance 1 for Predicate Logic|Hilbert Proof System Instance 1 for Predicate ... | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Forward Implication | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2/Forward_Implication | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)",
"Hilbert Proof System Instance 1 for Predicate Logic"
] | [] | [
"Definition:Hilbert Proof System/Predicate Logic/Instance 1",
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Provable Consequence of Theorems is Theorem",
"Deduction Theorem for Hilbert Proof System for Predicate Logic",
"Rule of Transposition",
"Rule of Transposition/Variant 2",
"Hyp... |
proofwiki-19943 | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Reverse Implication | :$\vdash \paren{ \exists x: \neg \map P x } \implies \neg \paren {\forall x: \map P x}$ | {{WIP}} | :$\vdash \paren{ \exists x: \neg \map P x } \implies \neg \paren {\forall x: \map P x}$ | {{WIP}} | De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Reverse Implication | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Universality/Formulation_2/Reverse_Implication | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [] |
proofwiki-19944 | Continuous Mapping from Compact Space to Hausdorff Space is Closed Mapping | Let $T_1 = \struct {S_1, \tau_1}$ be a compact topological space.
Let $T_2 = \struct {S_2, \tau_2}$ be a $T_2$ (Hausdorff) space.
Let $f: T_1 \to T_2$ be a continuous mapping.
Then $f$ is a closed mapping. | Let $C$ be a closed subspace of $T_1$.
By Closed Subspace of Compact Space is Compact, $C$ is compact.
By Continuous Image of Compact Space is Compact, $f \sqbrk C$ is compact in $T_2$.
By Compact Subspace of Hausdorff Space is Closed, $f \sqbrk C$ is closed in $T_2$.
The result follows by definition of closed mapping.... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $T_2 = \struct {S_2, \tau_2}$ be a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]].
Let $f: T_1 \to T_2$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Then $f$ is ... | Let $C$ be a [[Definition:Closed Set (Topology)|closed subspace]] of $T_1$.
By [[Closed Subspace of Compact Space is Compact]], $C$ is [[Definition:Compact Topological Space|compact]].
By [[Continuous Image of Compact Space is Compact]], $f \sqbrk C$ is [[Definition:Compact Topological Space|compact]] in $T_2$.
By [... | Continuous Mapping from Compact Space to Hausdorff Space is Closed Mapping | https://proofwiki.org/wiki/Continuous_Mapping_from_Compact_Space_to_Hausdorff_Space_is_Closed_Mapping | https://proofwiki.org/wiki/Continuous_Mapping_from_Compact_Space_to_Hausdorff_Space_is_Closed_Mapping | [
"Compact Topological Spaces",
"Hausdorff Spaces",
"Closed Mappings"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Closed Mapping"
] | [
"Definition:Closed Set/Topology",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Topological Space",
"Continuous Image of Compact Space is Compact",
"Definition:Compact Topological Space",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set/Topology",
"Definitio... |
proofwiki-19945 | Lamé's Theorem/Least Absolute Remainder | Let $a, b \in \Z_{>0}$ be (strictly) positive integers.
Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation.
Let the Euclidean Algorithm: Least Absolute Remainder variant be employed to find the GCD of $a$ and $b$.
Then it will in general take fewer integer divisions t... | === Lemma===
{{:Lamé's Theorem/Least Absolute Remainder/Lemma}}{{qed|lemma}}
{{WLOG}} suppose $a \ge b$.
Then $\min \set {c, d}$ is the number of digits in $b$.
By Number of Digits in Number, we have:
:$\min \set {c, d} = \floor {\log b} + 1$
{{AimForCont}} it takes at least $3 \paren {\floor {\log b} + 1}$ cycles aro... | Let $a, b \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $c$ and $d$ be the number of [[Definition:Digit|digits]] in $a$ and $b$ respectively when expressed in [[Definition:Decimal Notation|decimal notation]].
Let the [[Euclidean Algorithm/Least Absolute Remainder|Euclidean... | === [[Lamé's Theorem/Least Absolute Remainder/Lemma|Lemma]]===
{{:Lamé's Theorem/Least Absolute Remainder/Lemma}}{{qed|lemma}}
{{WLOG}} suppose $a \ge b$.
Then $\min \set {c, d}$ is the number of [[Definition:Digit|digits]] in $b$.
By [[Number of Digits in Number]], we have:
:$\min \set {c, d} = \floor {\log b} + ... | Lamé's Theorem/Least Absolute Remainder | https://proofwiki.org/wiki/Lamé's_Theorem/Least_Absolute_Remainder | https://proofwiki.org/wiki/Lamé's_Theorem/Least_Absolute_Remainder | [
"Euclidean Algorithm",
"Lamé's Theorem"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Digit",
"Definition:Decimal Notation",
"Euclidean Algorithm/Least Absolute Remainder",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Division",
"Euclidean Algorithm",
"Definition:Integer Division"
] | [
"Lamé's Theorem/Least Absolute Remainder/Lemma",
"Definition:Digit",
"Number of Digits in Number",
"Euclidean Algorithm/Least Absolute Remainder",
"Lamé's Theorem/Least Absolute Remainder/Lemma",
"Lower Bound of Pell Number",
"Definition:Contradiction",
"Logarithm of Power/General Logarithm",
"Defin... |
proofwiki-19946 | GCD of Integers with Common Divisor/Corollary | Let $a, b \in \Z$ be integers such that not both $a = 0$ and $b = 0$.
Let $k \in \Z_{\ne 0}$ be a non-zero integer.
Then:
:$\gcd \set {k a, k b} = \size k \gcd \set {a, b}$
where $\gcd$ denotes the greatest common divisor. | From GCD of Integers with Common Divisor the case where $k > 0$ has been demonstrated.
It remains to demonstrate the case where $k < 0$.
Indeed:
:$-k = \size k > 0$
and so:
{{begin-eqn}}
{{eqn | l = \gcd \set {a k, b k}
| r = \gcd \set {-a k, -b k}
| c =
}}
{{eqn | r = \gcd \set {a \size k, b \size k}
... | Let $a, b \in \Z$ be [[Definition:Integer|integers]] such that not both $a = 0$ and $b = 0$.
Let $k \in \Z_{\ne 0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Integer|integer]].
Then:
:$\gcd \set {k a, k b} = \size k \gcd \set {a, b}$
where $\gcd$ denotes the [[Definition:Greatest Common Divisor of Inte... | From [[GCD of Integers with Common Divisor]] the case where $k > 0$ has been demonstrated.
It remains to demonstrate the case where $k < 0$.
Indeed:
:$-k = \size k > 0$
and so:
{{begin-eqn}}
{{eqn | l = \gcd \set {a k, b k}
| r = \gcd \set {-a k, -b k}
| c =
}}
{{eqn | r = \gcd \set {a \size k, b \siz... | GCD of Integers with Common Divisor/Corollary | https://proofwiki.org/wiki/GCD_of_Integers_with_Common_Divisor/Corollary | https://proofwiki.org/wiki/GCD_of_Integers_with_Common_Divisor/Corollary | [
"GCD of Integers with Common Divisor"
] | [
"Definition:Integer",
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Greatest Common Divisor/Integers"
] | [
"GCD of Integers with Common Divisor"
] |
proofwiki-19947 | Equivalence of Definitions of Non-Archimedean Vector Space Norm | Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.
Let $X$ be a vector space over $R$, with zero $0_X$.
{{TFAE|def = Non-Archimedean Vector Space Norm}}
=== Definition 1 ===
{{:Definition:Non-Archimedean/Norm (Vector Space)/Definition 1}}
=== Definition 2 ===
{{:Definition:Non-Archimedean/N... | === Definition 1 implies Definition 2 ===
Let $\norm {\,\cdot\,} : X \to \R_{\ge 0}$ be a norm on a division ring satisfying:
{{begin-axiom}}
{{axiom | n = \text N 4
| lc= Ultrametric Inequality:
| q = \forall x, y \in X
| ml= \norm {x + y}
| mo= \le
| mr= \max \set {\norm x, \no... | Let $\struct {R, +, \circ}$ be a [[Definition:Division Ring|division ring]] with [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}_R$.
Let $X$ be a [[Definition:Vector Space over Division Ring|vector space]] over $R$, with [[Definition:Zero Vector|zero]] $0_X$.
{{TFAE|def = Non-Archimedean Vector Space No... | === Definition 1 implies Definition 2 ===
Let $\norm {\,\cdot\,} : X \to \R_{\ge 0}$ 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 X
| ml= \norm {x + y}
| mo= \l... | Equivalence of Definitions of Non-Archimedean Vector Space Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Non-Archimedean_Vector_Space_Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Non-Archimedean_Vector_Space_Norm | [
"Norm Theory"
] | [
"Definition:Division Ring",
"Definition:Norm/Division Ring",
"Definition:Vector Space/Division Ring",
"Definition:Zero Vector",
"Definition:Non-Archimedean/Norm (Vector Space)/Definition 1",
"Definition:Non-Archimedean/Norm (Vector Space)/Definition 2"
] | [
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring"
] |
proofwiki-19948 | Path as Parameterization of Contour/Corollary 1 | If $\gamma$ is a closed path, then $C$ is a closed contour. | By definition of closed path, we have
:$\map \gamma a = \map {\gamma_1} {a_0} = \map {\gamma_n} {a_n}$
so:
:$C_1$ has start point $\map \gamma a$
and:
:$C_n$ has end point $\map \gamma a$.
By definition, it follows that $C$ is a closed contour.
{{qed}}
Category:Path as Parameterization of Contour
syjz8x3d90onovtv1n8rz4... | If $\gamma$ is a [[Definition:Closed Path (Topology)|closed path]], then $C$ is a [[Definition:Closed Contour (Complex Plane)|closed contour]]. | By definition of [[Definition:Closed Path (Topology)|closed path]], we have
:$\map \gamma a = \map {\gamma_1} {a_0} = \map {\gamma_n} {a_n}$
so:
:$C_1$ has [[Definition:Start Point of Contour (Complex Plane)|start point]] $\map \gamma a$
and:
:$C_n$ has [[Definition:End Point of Contour (Complex Plane)|end point]]... | Path as Parameterization of Contour/Corollary 1 | https://proofwiki.org/wiki/Path_as_Parameterization_of_Contour/Corollary_1 | https://proofwiki.org/wiki/Path_as_Parameterization_of_Contour/Corollary_1 | [
"Path as Parameterization of Contour"
] | [
"Definition:Loop (Topology)",
"Definition:Contour/Closed/Complex Plane"
] | [
"Definition:Loop (Topology)",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Category:Path as Parameterization of Contour"
] |
proofwiki-19949 | Path as Parameterization of Contour/Corollary 2 | If $\gamma$ is a Jordan arc, then $C$ is a simple contour, and if $\gamma$ is a Jordan curve, then $C$ is a simple closed contour. | Let $k_1, k_2 \in \set {1, \ldots, n}$, and $t_1 \in \hointr {a_{k_1 - 1} } {a_{k_1} }, t_2 \in \hointr {a_{k_2 - 1} } {a_{k_2} }$.
Then by the definition of Jordan arc, or Jordan curve:
:$\map \gamma {t_1} \ne \map \gamma {t_2}$
so:
:$\map {\gamma_{k_1} } {t_1} \ne \map {\gamma_{k_2} } {t_2}$
Let instead:
:$k \in \set... | If $\gamma$ is a [[Definition:Jordan Arc|Jordan arc]], then $C$ is a [[Definition:Simple Contour (Complex Plane)|simple contour]], and if $\gamma$ is a [[Definition:Jordan Curve|Jordan curve]], then $C$ is a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]... | Let $k_1, k_2 \in \set {1, \ldots, n}$, and $t_1 \in \hointr {a_{k_1 - 1} } {a_{k_1} }, t_2 \in \hointr {a_{k_2 - 1} } {a_{k_2} }$.
Then by the definition of [[Definition:Jordan Arc|Jordan arc]], or [[Definition:Jordan Curve|Jordan curve]]:
:$\map \gamma {t_1} \ne \map \gamma {t_2}$
so:
:$\map {\gamma_{k_1} } {t_1}... | Path as Parameterization of Contour/Corollary 2 | https://proofwiki.org/wiki/Path_as_Parameterization_of_Contour/Corollary_2 | https://proofwiki.org/wiki/Path_as_Parameterization_of_Contour/Corollary_2 | [
"Path as Parameterization of Contour"
] | [
"Definition:Jordan Arc",
"Definition:Contour/Simple/Complex Plane",
"Definition:Jordan Curve",
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane"
] | [
"Definition:Jordan Arc",
"Definition:Jordan Curve",
"Definition:Jordan Arc",
"Definition:Jordan Curve",
"Definition:Contour/Simple/Complex Plane",
"Definition:Jordan Curve",
"Definition:Loop (Topology)",
"Path as Parameterization of Contour/Corollary 1",
"Definition:Jordan Curve",
"Definition:Cont... |
proofwiki-19950 | Volume of Paraboloid | The volume of paraboloid is half the volume of its circumscribing cylinder. | {{MissingLinks}}
420pxright
Consider a cylinder of radius $r$ and height $h$, circumscribing a paraboloid $y = h \paren {\dfrac x r}^2$ whose apex is at the center of the bottom base of the cylinder and whose base is the top base of the cylinder.
Also consider the paraboloid $y = h - h \paren {\dfrac x r}^2$, with equa... | The [[Definition:Volume|volume]] of [[Definition:Paraboloid|paraboloid]] is [[Definition:Half|half]] the [[Definition:Volume|volume]] of its [[Definition:Circumscribe|circumscribing]] [[Definition:Right Circular Cylinder|cylinder]]. | {{MissingLinks}}
[[File:Cavalieri's principle - Volume of paraboloid.gif|420px|right]]
Consider a [[Definition:Right Circular Cylinder|cylinder]] of radius $r$ and height $h$, circumscribing a [[Definition:Paraboloid|paraboloid]] $y = h \paren {\dfrac x r}^2$ whose apex is at the center of the bottom base of the [[De... | Volume of Paraboloid | https://proofwiki.org/wiki/Volume_of_Paraboloid | https://proofwiki.org/wiki/Volume_of_Paraboloid | [
"Paraboloids",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Paraboloid",
"Definition:Half",
"Definition:Volume",
"Definition:Circumscribe",
"Definition:Right Circular Cylinder"
] | [
"File:Cavalieri's principle - Volume of paraboloid.gif",
"Definition:Right Circular Cylinder",
"Definition:Paraboloid",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cyl... |
proofwiki-19951 | Quotient Mapping equals Surjective Identification Mapping | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a mapping.
Then $f$ is a quotient mapping, {{Iff}}:
:$f$ is surjective, and $\tau_2$ is the identification topology on $S_2$ with respect to $f$ and $T_1$. | === Sufficient condition ===
Let $f$ be surjective, and $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $T_1$.
Identification Mapping is Continuous shows that $f$ is continuous.
Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is open in $T_1$.
By definition of identification topology, it foll... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Then $f$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]], {{Iff}}:
:$f$ is [[Definition:Surjection|surjective]], and ... | === Sufficient condition ===
Let $f$ be [[Definition:Surjection|surjective]], and $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $T_1$.
[[Identification Mapping is Continuous]] shows that $f$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Let $U \subseteq S_2$ such that $f^{-1}... | Quotient Mapping equals Surjective Identification Mapping | https://proofwiki.org/wiki/Quotient_Mapping_equals_Surjective_Identification_Mapping | https://proofwiki.org/wiki/Quotient_Mapping_equals_Surjective_Identification_Mapping | [
"Quotient Mappings (Topology)",
"Identification Topology"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Quotient Mapping (Topology)",
"Definition:Surjection",
"Definition:Identification Topology"
] | [
"Definition:Surjection",
"Identification Mapping is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Identification Topology",
"Definition:Open Set/Topology",
"Definition:Quotient Mapping (Topology)",
"Definition:Surjection",
"Definition:Identifica... |
proofwiki-19952 | Quotient Topology is Topology | Let $\struct {S, \tau}$ be a topological space.
Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.
Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$.
Let $\tau_\RR$ be the quotient topology on $S / \RR$ by $q_\RR$.
Then $\tau_\RR$ is a topology on $S$. | By definition of quotient topology, $\tau_\RR$ is the identification topology on $S / \RR$ with respect to $q_\RR$ and $\struct {S, \tau}$.
Identification Topology is Topology shows that $\tau_\RR$ is a topology on $S / \RR$.
{{qed}} | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\RR \subseteq S \times S$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $q_\RR: S \to S / \RR$ be the [[Definition:Quotient Mapping|quotient mapping]] induced by $\RR$.
Let $\tau_\RR$ be the [[Definitio... | By definition of [[Definition:Quotient Topology/Definition 1|quotient topology]], $\tau_\RR$ is the [[Definition:Identification Topology|identification topology]] on $S / \RR$ with respect to $q_\RR$ and $\struct {S, \tau}$.
[[Identification Topology is Topology]] shows that $\tau_\RR$ is a [[Definition:Topology|topol... | Quotient Topology is Topology | https://proofwiki.org/wiki/Quotient_Topology_is_Topology | https://proofwiki.org/wiki/Quotient_Topology_is_Topology | [
"Quotient Topologies"
] | [
"Definition:Topological Space",
"Definition:Equivalence Relation",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Topology"
] | [
"Definition:Quotient Topology/Definition 1",
"Definition:Identification Topology",
"Identification Topology is Topology",
"Definition:Topology"
] |
proofwiki-19953 | Quotient Mapping Induces Homeomorphism between Quotient Space and Image | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a quotient mapping.
Let $\RR_f \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $f$:
:$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$
Let $q_{\RR_f}: S_1 \to S_1 / \RR_f$ be the quotient... | Quotient Mapping equals Surjective Identification Mapping shows that $\tau_2$ is the identification topology on $S_2$ with respect to $f$ and $\struct { S_1, \tau_1}$.
Identification Topology equals Quotient Topology on Induced Equivalence shows that $\struct {S_1 / \RR_f, \tau_{\RR_f} }$ and $\struct {S_2, \tau_2}$ ar... | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Quotient Mapping (Topology)|quotient mapping]].
Let $\RR_f \subseteq S_1 \times S_1$ be the [[Definition:Equivalence Relation Induced by Mapping|equivalence]] on $S_1$... | [[Quotient Mapping equals Surjective Identification Mapping]] shows that $\tau_2$ is the [[Definition:Identification Topology|identification topology]] on $S_2$ with respect to $f$ and $\struct { S_1, \tau_1}$.
[[Identification Topology equals Quotient Topology on Induced Equivalence]] shows that $\struct {S_1 / \RR_f... | Quotient Mapping Induces Homeomorphism between Quotient Space and Image | https://proofwiki.org/wiki/Quotient_Mapping_Induces_Homeomorphism_between_Quotient_Space_and_Image | https://proofwiki.org/wiki/Quotient_Mapping_Induces_Homeomorphism_between_Quotient_Space_and_Image | [
"Quotient Mappings (Topology)",
"Quotient Spaces (Topology)"
] | [
"Definition:Topological Space",
"Definition:Quotient Mapping (Topology)",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Homeomorphism",
"Definition:Homeomorphism",
"Definition:Quotient Topology/Quotient Space"
] | [
"Quotient Mapping equals Surjective Identification Mapping",
"Definition:Identification Topology",
"Identification Topology equals Quotient Topology on Induced Equivalence",
"Definition:Homeomorphism",
"Identification Topology equals Quotient Topology on Induced Equivalence",
"Definition:Homeomorphism",
... |
proofwiki-19954 | Continuous Closed Surjective Mapping is Quotient Mapping | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a continuous closed surjective mapping.
Then $f$ is a quotient mapping. | Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is open in $T_1$.
For $X \subseteq S$, let $\relcomp S X$ denote the relative complement of $X$ is $S$.
By definition of closed set, $\relcomp {S_1} {f^{-1} \sqbrk U}$ is closed in $T_1$.
By definition of closed mapping, $f \sqbrk {\relcomp {S_1} {f^{-1} \sqbrk U} }$ is... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Closed Mapping|closed]] [[Definition:Surjection|surjective mapping]].
Then $f$ is a [[Definition:Q... | Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is [[Definition:Open Set (Topology)|open]] in $T_1$.
For $X \subseteq S$, let $\relcomp S X$ denote the [[Definition:Relative Complement|relative complement]] of $X$ is $S$.
By definition of [[Definition:Closed Set (Topology)|closed set]], $\relcomp {S_1} {f^{-1} \sqb... | Continuous Closed Surjective Mapping is Quotient Mapping | https://proofwiki.org/wiki/Continuous_Closed_Surjective_Mapping_is_Quotient_Mapping | https://proofwiki.org/wiki/Continuous_Closed_Surjective_Mapping_is_Quotient_Mapping | [
"Closed Mappings",
"Surjections",
"Quotient Mappings (Topology)",
"Quotient Topologies"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Closed Mapping",
"Definition:Surjection",
"Definition:Quotient Mapping (Topology)"
] | [
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Mapping",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Complement of Preimage equals Preima... |
proofwiki-19955 | Upper Bound for Lowest Common Multiple | Let $a, b \in \Z$ be integers such that $a b \ne 0$.
Then:
:$\lcm \set {a, b} \le \size {a b}$
where:
:$\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$ | By Product of GCD and LCM:
:$\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$
where:
:$\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
By Existence of Greatest Common Divisor $\gcd \set {a, b}$ exists.
By definition of GCD, $\gcd \set {a, b} \in \Z_{>0}$.
Hence the result.
{{qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]] such that $a b \ne 0$.
Then:
:$\lcm \set {a, b} \le \size {a b}$
where:
:$\lcm \set {a, b}$ denotes the [[Definition:Lowest Common Multiple of Integers|lowest common multiple]] of $a$ and $b$ | By [[Product of GCD and LCM]]:
:$\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$
where:
:$\gcd \set {a, b}$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
By [[Existence of Greatest Common Divisor]] $\gcd \set {a, b}$ exists.
By definition of [[Definitio... | Upper Bound for Lowest Common Multiple | https://proofwiki.org/wiki/Upper_Bound_for_Lowest_Common_Multiple | https://proofwiki.org/wiki/Upper_Bound_for_Lowest_Common_Multiple | [
"Lowest Common Multiple"
] | [
"Definition:Integer",
"Definition:Lowest Common Multiple/Integers"
] | [
"Product of GCD and LCM",
"Definition:Greatest Common Divisor/Integers",
"Existence of Greatest Common Divisor",
"Definition:Greatest Common Divisor/Integers"
] |
proofwiki-19956 | Dilation of Convex Set in Vector Space is Convex | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a vector space over $\Bbb F$.
Let $C \subseteq X$ be a convex subset of $X$.
Let $\alpha \in \Bbb F$.
Then $\alpha C$ is convex. | Consider first the case $\alpha = 0$.
We then have $\alpha C = \set { {\mathbf 0}_X}$.
This is convex by Singleton is Convex Set.
Now consider the case $\alpha \ne 0$.
Let $u, v \in \alpha C$ and $t \in \closedint 0 1$.
Then there exists $x, y \in C$ such that $u = \alpha x$ and $v = \alpha y$.
Since $C$ is convex, w... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\Bbb F$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Let $\alpha \in \Bbb F$.
Then $\alpha C$ is [[Definition:Convex Set (Vector Space)|convex]]. | Consider first the case $\alpha = 0$.
We then have $\alpha C = \set { {\mathbf 0}_X}$.
This is [[Definition:Convex Set (Vector Space)|convex]] by [[Singleton is Convex Set]].
Now consider the case $\alpha \ne 0$.
Let $u, v \in \alpha C$ and $t \in \closedint 0 1$.
Then there exists $x, y \in C$ such that $u = \... | Dilation of Convex Set in Vector Space is Convex | https://proofwiki.org/wiki/Dilation_of_Convex_Set_in_Vector_Space_is_Convex | https://proofwiki.org/wiki/Dilation_of_Convex_Set_in_Vector_Space_is_Convex | [
"Dilations of Subsets of Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Set (Vector Space)",
"Singleton is Convex Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Category:Dilations of Subsets of Vector Spaces",
"Category:Convex Sets (Vector Spaces)"
] |
proofwiki-19957 | Continuous Open Surjective Mapping is Quotient Mapping | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a continuous open surjective mapping.
Then $f$ is a quotient mapping. | Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is open in $T_1$.
By definition of open mapping:
:$f \sqbrk {f^{-1} \sqbrk U}$ is open in $T_2$
From {{Corollary|Image of Preimage under Mapping}}:
:$f \sqbrk {f^{-1} \sqbrk U} = U$
It follows that $U$ is open in $T_2$.
By definition of quotient mapping, it follows that... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Open Mapping|open]] [[Definition:Surjection|surjective mapping]].
Then $f$ is a [[Definition:Quoti... | Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is [[Definition:Open Set (Topology)|open]] in $T_1$.
By definition of [[Definition:Open Mapping|open mapping]]:
:$f \sqbrk {f^{-1} \sqbrk U}$ is [[Definition:Open Set (Topology)|open]] in $T_2$
From {{Corollary|Image of Preimage under Mapping}}:
:$f \sqbrk {f^{-1} \sq... | Continuous Open Surjective Mapping is Quotient Mapping | https://proofwiki.org/wiki/Continuous_Open_Surjective_Mapping_is_Quotient_Mapping | https://proofwiki.org/wiki/Continuous_Open_Surjective_Mapping_is_Quotient_Mapping | [
"Open Mappings",
"Surjections",
"Quotient Mappings (Topology)",
"Quotient Topologies"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Mapping",
"Definition:Surjection",
"Definition:Quotient Mapping (Topology)"
] | [
"Definition:Open Set/Topology",
"Definition:Open Mapping",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Quotient Mapping (Topology)",
"Definition:Quotient Mapping (Topology)"
] |
proofwiki-19958 | Composite of Quotient Mappings in Topology is Quotient Mapping | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be topological spaces.
Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be quotient mappings.
Then $g \circ f : S_1 \to S_3$ is a quotient mapping. | Composite of Surjections is Surjection shows that $g \circ f$ is surjective.
Composite of Continuous Mappings is Continuous shows that $g \circ f$ is continuous.
Let $U \subseteq S_3$ such that $\paren {g \circ f}^{-1} \sqbrk U$ is open in $T_1$.
By definition of quotient mapping:
:$f \sqbrk {\paren {g \circ f}^{-1} \s... | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be [[Definition:Quotient Mapping (Topology)|quotient mappings]].
Then $g \circ f : S_1 \to S_3$ is a [[Definition:Quotient ... | [[Composite of Surjections is Surjection]] shows that $g \circ f$ is [[Definition:Surjection|surjective]].
[[Composite of Continuous Mappings is Continuous]] shows that $g \circ f$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Let $U \subseteq S_3$ such that $\paren {g \circ f}^{-1} \sqbrk U$ is [[Defin... | Composite of Quotient Mappings in Topology is Quotient Mapping | https://proofwiki.org/wiki/Composite_of_Quotient_Mappings_in_Topology_is_Quotient_Mapping | https://proofwiki.org/wiki/Composite_of_Quotient_Mappings_in_Topology_is_Quotient_Mapping | [
"Quotient Mappings (Topology)",
"Quotient Topologies"
] | [
"Definition:Topological Space",
"Definition:Quotient Mapping (Topology)",
"Definition:Quotient Mapping (Topology)"
] | [
"Composite of Surjections is Surjection",
"Definition:Surjection",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Quotient Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Quotient Mapping (Topol... |
proofwiki-19959 | Quotient Mapping and Continuous Mapping Induces Continuous Mapping | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be topological spaces.
Let $p: S_1 \to S_2$ be a quotient mapping.
Let $g: S_2 \to S_3$ be a mapping such that for all $s_1, s_2 \in S_1$ with $\map p {s_1} = \map p {s_2}$, we have $\map g {s_1} = \map g {s_2}$.
Then $g$ in... | For all $t \in S_2$ we can find $s \in S_1$ with $\map p s = t$, as $p$ is surjective.
Define $\map f t := \map g s$.
For all $s' \in S_1$ with $\map p {s'} = t$, we have $\map g s = \map g {s'}$ {{hypothesis}}.
Hence $f$ is well-defined.
It follows that $\map g s = \map {f \circ p} s$.
Suppose $f$ is continuous.
As $p... | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be [[Definition:Topological Space|topological spaces]].
Let $p: S_1 \to S_2$ be a [[Definition:Quotient Mapping (Topology)|quotient mapping]].
Let $g: S_2 \to S_3$ be a [[Definition:Mapping|mapping]] such that for all $s_1... | For all $t \in S_2$ we can find $s \in S_1$ with $\map p s = t$, as $p$ is [[Definition:Surjection|surjective]].
Define $\map f t := \map g s$.
For all $s' \in S_1$ with $\map p {s'} = t$, we have $\map g s = \map g {s'}$ {{hypothesis}}.
Hence $f$ is [[Definition:Well-Defined Mapping|well-defined]].
It follows that... | Quotient Mapping and Continuous Mapping Induces Continuous Mapping | https://proofwiki.org/wiki/Quotient_Mapping_and_Continuous_Mapping_Induces_Continuous_Mapping | https://proofwiki.org/wiki/Quotient_Mapping_and_Continuous_Mapping_Induces_Continuous_Mapping | [
"Quotient Mappings (Topology)"
] | [
"Definition:Topological Space",
"Definition:Quotient Mapping (Topology)",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping"
] | [
"Definition:Surjection",
"Definition:Well-Defined/Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Quotient Mapping (Topology)",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Contin... |
proofwiki-19960 | Set of Integers with GCD of 1 are not necessarily Pairwise Coprime | Let $S$ be a set of integers such that $S$ has more than $2$ elements:
:$S = \set {s_1, s_2, \ldots, s_n}$
Let:
:$\map \gcd S = 1$
where $\gcd$ denotes the GCD of $S$.
Then it is not necessarily the case that there exist a pair of elements of $S$ which are themselves pairwise coprime:
:$\exists i, j \in \set {1, 2, \ld... | Proof by Counterexample
Let $S = \set {6, 10, 15}$.
We have:
{{begin-eqn}}
{{eqn | l = \gcd \set {6, 10}
| r = 2
| c =
}}
{{eqn | l = \gcd \set {6, 15}
| r = 3
| c =
}}
{{eqn | l = \gcd \set {10, 15}
| r = 5
| c =
}}
{{end-eqn}}
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Set|set]] of [[Definition:Integer|integers]] such that $S$ has more than $2$ [[Definition:Element|elements]]:
:$S = \set {s_1, s_2, \ldots, s_n}$
Let:
:$\map \gcd S = 1$
where $\gcd$ denotes the [[Definition:Greatest Common Divisor of Set of Integers|GCD]] of $S$.
Then it is not necessaril... | [[Proof by Counterexample]]
Let $S = \set {6, 10, 15}$.
We have:
{{begin-eqn}}
{{eqn | l = \gcd \set {6, 10}
| r = 2
| c =
}}
{{eqn | l = \gcd \set {6, 15}
| r = 3
| c =
}}
{{eqn | l = \gcd \set {10, 15}
| r = 5
| c =
}}
{{end-eqn}}
Hence the result.
{{qed}} | Set of Integers with GCD of 1 are not necessarily Pairwise Coprime | https://proofwiki.org/wiki/Set_of_Integers_with_GCD_of_1_are_not_necessarily_Pairwise_Coprime | https://proofwiki.org/wiki/Set_of_Integers_with_GCD_of_1_are_not_necessarily_Pairwise_Coprime | [
"Greatest Common Divisor",
"Coprime Integers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Element",
"Definition:Greatest Common Divisor/Integers/General Definition",
"Definition:Doubleton",
"Definition:Element",
"Definition:Pairwise Coprime/Integers"
] | [
"Proof by Counterexample"
] |
proofwiki-19961 | Quotient Mapping and Continuous Mapping Induces Continuous Mapping/Corollary | The induced mapping $f$ is a quotient mapping, {{iff}} $g$ is a quotient mapping. | Quotient Mapping and Continuous Mapping Induces Continuous Mapping shows that $f$ is well-defined.
Suppose $f$ is a quotient mapping.
Composite of Quotient Mappings in Topology is Quotient Mapping shows that $g = f \circ p$ is a quotient mapping.
Suppose $g$ is a quotient mapping.
It follows that $g$ is surjective.
Tha... | The induced [[Definition:Mapping|mapping]] $f$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]], {{iff}} $g$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]]. | [[Quotient Mapping and Continuous Mapping Induces Continuous Mapping]] shows that $f$ is well-defined.
Suppose $f$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]].
[[Composite of Quotient Mappings in Topology is Quotient Mapping]] shows that $g = f \circ p$ is a [[Definition:Quotient Mapping (Topolog... | Quotient Mapping and Continuous Mapping Induces Continuous Mapping/Corollary | https://proofwiki.org/wiki/Quotient_Mapping_and_Continuous_Mapping_Induces_Continuous_Mapping/Corollary | https://proofwiki.org/wiki/Quotient_Mapping_and_Continuous_Mapping_Induces_Continuous_Mapping/Corollary | [
"Quotient Mappings (Topology)"
] | [
"Definition:Mapping",
"Definition:Quotient Mapping (Topology)",
"Definition:Quotient Mapping (Topology)"
] | [
"Quotient Mapping and Continuous Mapping Induces Continuous Mapping",
"Definition:Quotient Mapping (Topology)",
"Composite of Quotient Mappings in Topology is Quotient Mapping",
"Definition:Quotient Mapping (Topology)",
"Definition:Quotient Mapping (Topology)",
"Definition:Surjection",
"Definition:Surje... |
proofwiki-19962 | Injective Quotient Mapping Equals Homeomorphism | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a mapping.
Then:
:$f$ is an injective quotient mapping
{{iff}}:
:$f$ is a homeomorphism. | === Sufficient condition ===
Suppose $f$ is an injective quotient mapping.
By definition of quotient mapping, $f$ is surjective.
Mapping is Injection and Surjection iff Inverse is Mapping shows that $f$ has an inverse $f^{-1}$.
To show continuity of $f^{-1}$, let $U \subseteq S_1$ be open in $T_1$.
As $f$ is bijective,... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Then:
:$f$ is an [[Definition:Injection|injective]] [[Definition:Quotient Mapping (Topology)|quotient mapping]]
{{iff}}:
:$f$ is a [[De... | === Sufficient condition ===
Suppose $f$ is an [[Definition:Injection|injective]] [[Definition:Quotient Mapping (Topology)|quotient mapping]].
By definition of [[Definition:Quotient Mapping (Topology)|quotient mapping]], $f$ is [[Definition:Surjection|surjective]].
[[Mapping is Injection and Surjection iff Inverse i... | Injective Quotient Mapping Equals Homeomorphism | https://proofwiki.org/wiki/Injective_Quotient_Mapping_Equals_Homeomorphism | https://proofwiki.org/wiki/Injective_Quotient_Mapping_Equals_Homeomorphism | [
"Quotient Mappings (Topology)",
"Injections",
"Quotient Topologies",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Injection",
"Definition:Quotient Mapping (Topology)",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Injection",
"Definition:Quotient Mapping (Topology)",
"Definition:Quotient Mapping (Topology)",
"Definition:Surjection",
"Mapping is Injection and Surjection iff Inverse is Mapping",
"Definition:Inverse of Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology"... |
proofwiki-19963 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step | Let the following hold:
:For all $j \le k$, a tree of order $j$ is of size $j - 1$.
Then this holds:
:A tree of order $k + 1$ is of size $k$. | Let $T_{k + 1}$ be an arbitrary tree with $k + 1$ nodes.
Take any node $v$ of $T_{k + 1}$ of degree $1$.
Such a node exists from Finite Tree has Leaf Nodes.
Consider $T_k$, the subgraph of $T_{k + 1}$ created by removing $v$ and the edge connecting it to the rest of the tree.
By Connected Subgraph of Tree is Tree, $T_k... | Let the following hold:
:For all $j \le k$, a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $j$ is of [[Definition:Size of Graph|size]] $j - 1$.
Then this holds:
:A [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $k + 1$ is of [[Definition:Size of Graph... | Let $T_{k + 1}$ be an arbitrary [[Definition:Tree (Graph Theory)|tree]] with $k + 1$ [[Definition:Node of Tree|nodes]].
Take any [[Definition:Node of Tree|node]] $v$ of $T_{k + 1}$ of [[Definition:Degree of Vertex|degree]] $1$.
Such a node exists from [[Finite Tree has Leaf Nodes]].
Consider $T_k$, the [[Definition:... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step/Proof 1 | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step/Proof_1 | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Tree (Graph Theory)/Node",
"Definition:Degree of Vertex",
"Finite Tree has Leaf Nodes",
"Definition:Subgraph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Tree (Graph Theory)",
"Connected Subgraph of Tree is... |
proofwiki-19964 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step | Let the following hold:
:For all $j \le k$, a tree of order $j$ is of size $j - 1$.
Then this holds:
:A tree of order $k + 1$ is of size $k$. | Let $T_{k + 1}$ be an arbitrary tree with $k + 1$ nodes.
Remove any edge $e$ of $T_{k + 1}$.
By definition of tree $T_{k + 1}$ has no circuits.
Therefore from Condition for Edge to be Bridge it follows that $e$ must be a bridge.
So removing $e$ disconnects $T_{k + 1}$ into two trees $T_1$ and $T_2$, with $k_1$ and $k_2... | Let the following hold:
:For all $j \le k$, a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $j$ is of [[Definition:Size of Graph|size]] $j - 1$.
Then this holds:
:A [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $k + 1$ is of [[Definition:Size of Graph... | Let $T_{k + 1}$ be an arbitrary [[Definition:Tree (Graph Theory)|tree]] with $k + 1$ [[Definition:Node of Tree|nodes]].
Remove any [[Definition:Edge of Graph|edge]] $e$ of $T_{k + 1}$.
By definition of [[Definition:Tree (Graph Theory)|tree]] $T_{k + 1}$ has no [[Definition:Circuit (Graph Theory)|circuits]].
Therefor... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step/Proof 2 | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step/Proof_2 | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Tree (Graph Theory)",
"Definition:Circuit (Graph Theory)",
"Condition for Edge to be Bridge",
"Definition:Bridge (Graph Theory)",
"Definition:Connected (Graph Theory)/Graph/Disc... |
proofwiki-19965 | Semi-Inner Product with Zero Vector | Let $\struct {V, \innerprod \cdot \cdot}$ be a semi-inner product space.
Let $\mathbf 0_V$ be the zero vector of $V$.
Then for all $v \in V$:
:$\innerprod {\mathbf 0_V} v = \innerprod v {\mathbf 0_V} = 0$ | {{begin-eqn}}
{{eqn | l = \innerprod {\mathbf 0_V} v
| r = \innerprod {0 \cdot \mathbf 0_V} v
}}
{{eqn | r = 0 \cdot \innerprod {\mathbf 0_V} v
| c = Semi-Inner Product Axioms: Sesquilinearity
}}
{{eqn | r = 0
}}
{{end-eqn}}
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \innerprod v {\mathbf 0_V}
| r = \ove... | Let $\struct {V, \innerprod \cdot \cdot}$ be a [[Definition:Semi-Inner Product Space|semi-inner product space]].
Let $\mathbf 0_V$ be the [[Definition:Zero Vector|zero vector]] of $V$.
Then for all $v \in V$:
:$\innerprod {\mathbf 0_V} v = \innerprod v {\mathbf 0_V} = 0$ | {{begin-eqn}}
{{eqn | l = \innerprod {\mathbf 0_V} v
| r = \innerprod {0 \cdot \mathbf 0_V} v
}}
{{eqn | r = 0 \cdot \innerprod {\mathbf 0_V} v
| c = [[Definition:Semi-Inner Product|Semi-Inner Product Axioms]]: [[Definition:Sesquilinear Form|Sesquilinearity]]
}}
{{eqn | r = 0
}}
{{end-eqn}}
{{qed|lemma}}
{... | Semi-Inner Product with Zero Vector | https://proofwiki.org/wiki/Semi-Inner_Product_with_Zero_Vector | https://proofwiki.org/wiki/Semi-Inner_Product_with_Zero_Vector | [
"Semi-Inner Product Spaces"
] | [
"Definition:Semi-Inner Product Space",
"Definition:Zero Vector"
] | [
"Definition:Semi-Inner Product",
"Definition:Sesquilinear Form",
"Definition:Semi-Inner Product",
"Definition:Conjugate Symmetric Mapping"
] |
proofwiki-19966 | Continuous Surjection Induces Continuous Bijection from Quotient Space | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces.
Let $g: S_1 \to S_2$ be a $\tuple {\tau_1, \tau_2}$-continuous surjection.
Let $\RR_g \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $g$:
:$\tuple {s_1, s_2} \in \RR_g \iff \map g {s_1} = \map g {s_2}$
Let $q_{\RR_g}: S_1 \... | By definition of quotient topology, $q_{\RR_g}$ is a surjective identification mapping.
Quotient Mapping equals Surjective Identification Mapping shows that $q_{\RR_g}$ is a quotient mapping.
Quotient Mapping and Continuous Mapping Induces Continuous Mapping shows that $g$ induces a continuous mapping $f: S_1 / \RR_g \... | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $g: S_1 \to S_2$ be a [[Definition:Everywhere Continuous Mapping (Topology)|$\tuple {\tau_1, \tau_2}$-continuous]] [[Definition:Surjection|surjection]].
Let $\RR_g \subseteq S_1 \times S_1$ be the [[Def... | By definition of [[Definition:Quotient Topology|quotient topology]], $q_{\RR_g}$ is a [[Definition:Surjection|surjective]] [[Definition:Identification Mapping|identification mapping]].
[[Quotient Mapping equals Surjective Identification Mapping]] shows that $q_{\RR_g}$ is a [[Definition:Quotient Mapping (Topology)|quo... | Continuous Surjection Induces Continuous Bijection from Quotient Space | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space | [
"Continuous Surjection Induces Continuous Bijection from Quotient Space",
"Quotient Spaces (Topology)"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Quotient Mapping",
"Definition:Equivalence Relation",
"Definition:Quotient Topology",
"Definition:Quotient Topology/Quotient... | [
"Definition:Quotient Topology",
"Definition:Surjection",
"Definition:Identification Topology/Identification Mapping",
"Quotient Mapping equals Surjective Identification Mapping",
"Definition:Quotient Mapping (Topology)",
"Quotient Mapping and Continuous Mapping Induces Continuous Mapping",
"Definition:C... |
proofwiki-19967 | Vector Space of Sequences with Finite Support is Vector Space | Let $\struct {K, +_K, \circ_K}$ be a division ring.
Let $\struct{V, +, \cdot}$ be the vector space of sequences with finite support in $K$.
Then $V$ is a vector space over $K$. | Consider $V$ as a subset of the vector space of all mappings from $\N$ to $K$.
Let us apply the One-Step Vector Subspace Test.
Thus, let $\sequence{a_n}, \sequence{b_n} \in V$ be sequences over $K$ with finite support, and let $\lambda \in K$.
Then we need to show that $\sequence{ a_n + \lambda b_n }$ has finite suppor... | Let $\struct {K, +_K, \circ_K}$ be a [[Definition:Division Ring|division ring]].
Let $\struct{V, +, \cdot}$ be the [[Definition:Vector Space of Sequences with Finite Support|vector space of sequences with finite support]] in $K$.
Then $V$ is a [[Definition:Vector Space|vector space]] over $K$. | Consider $V$ as a [[Definition:Subset|subset]] of the [[Definition:Vector Space of All Mappings|vector space of all mappings]] from $\N$ to $K$.
Let us apply the [[One-Step Vector Subspace Test]].
Thus, let $\sequence{a_n}, \sequence{b_n} \in V$ be [[Definition:Sequence with Finite Support|sequences over $K$ with fi... | Vector Space of Sequences with Finite Support is Vector Space | https://proofwiki.org/wiki/Vector_Space_of_Sequences_with_Finite_Support_is_Vector_Space | https://proofwiki.org/wiki/Vector_Space_of_Sequences_with_Finite_Support_is_Vector_Space | [
"Examples of Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Vector Space of Sequences with Finite Support",
"Definition:Vector Space"
] | [
"Definition:Subset",
"Definition:Vector Space of All Mappings",
"One-Step Vector Subspace Test",
"Definition:Sequence with Finite Support",
"Definition:Sequence with Finite Support",
"Union of Finite Sets is Finite",
"Definition:Sequence with Finite Support",
"One-Step Vector Subspace Test",
"Catego... |
proofwiki-19968 | Expression for bilinear function | Let $f$ be a real function of two independent variables, $f \in \R \times \R \to \R$.
Then:
:$\map f {x, y}$ is a linear function of $x$ when $y$ is equal to a real constant
:$\map f {x, y}$ is a linear function of $y$ when $x$ is equal to a real constant
{{iff}} $f$ has the form:
:$\exists a, b, c, d \in \R: \forall {... | === Sufficient Condition ===
Let:
:$\map f {x, y}$ be a linear function of $x$ when $y$ is equal to a real constant
:$\map f {x, y}$ be a linear function of $y$ when $x$ is equal to a real constant
We need to show that:
:$\exists a, b, c, d \in \R: \forall {x, y} \in \R: \map f {x, y} = a x y + b x + c y + d$ | Let $f$ be a [[Definition:Real Function of Two Variables|real function]] of two [[Definition:Real Independent Variable|independent variables]], $f \in \R \times \R \to \R$.
Then:
:$\map f {x, y}$ is a [[Definition:Linear Real Function|linear function]] of $x$ when $y$ is equal to a [[Definition:Real Number|real]] [[D... | === Sufficient Condition ===
Let:
:$\map f {x, y}$ be a [[Definition:Linear Real Function|linear function]] of $x$ when $y$ is equal to a [[Definition:Real Number|real]] [[Definition:Constant|constant]]
:$\map f {x, y}$ be a [[Definition:Linear Real Function|linear function]] of $y$ when $x$ is equal to a [[Definitio... | Expression for bilinear function | https://proofwiki.org/wiki/Expression_for_bilinear_function | https://proofwiki.org/wiki/Expression_for_bilinear_function | [
"Linear Real Functions"
] | [
"Definition:Real Function/Two Variables",
"Definition:Independent Variable/Real Function",
"Definition:Linear Real Function",
"Definition:Real Number",
"Definition:Constant",
"Definition:Linear Real Function",
"Definition:Real Number",
"Definition:Constant"
] | [
"Definition:Linear Real Function",
"Definition:Real Number",
"Definition:Constant",
"Definition:Linear Real Function",
"Definition:Real Number",
"Definition:Constant",
"Definition:Linear Real Function",
"Definition:Real Number",
"Definition:Constant",
"Definition:Real Number",
"Definition:Consta... |
proofwiki-19969 | Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 1 | The induced mapping $f$ is a homeomorphism, {{iff}} $g$ is a quotient mapping. | Continuous Surjection Induces Continuous Bijection from Quotient Space shows that $g$ is well-defined.
Let $f$ be a homeomorphism.
Injective Quotient Mapping Equals Homeomorphism shows that $f$ is a quotient mapping.
Composite of Quotient Mappings in Topology is Quotient Mapping shows that $g = f \circ q_{\RR_g}$ is a ... | The induced [[Definition:Mapping|mapping]] $f$ is a [[Definition:Homeomorphism|homeomorphism]], {{iff}} $g$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]]. | [[Continuous Surjection Induces Continuous Bijection from Quotient Space]] shows that $g$ is well-defined.
Let $f$ be a [[Definition:Homeomorphism|homeomorphism]].
[[Injective Quotient Mapping Equals Homeomorphism]] shows that $f$ is a [[Definition:Quotient Mapping (Topology)|quotient mapping]].
[[Composite of Quoti... | Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 1 | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space/Corollary_1 | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space/Corollary_1 | [
"Continuous Surjection Induces Continuous Bijection from Quotient Space",
"Quotient Spaces (Topology)"
] | [
"Definition:Mapping",
"Definition:Homeomorphism",
"Definition:Quotient Mapping (Topology)"
] | [
"Continuous Surjection Induces Continuous Bijection from Quotient Space",
"Definition:Homeomorphism",
"Injective Quotient Mapping Equals Homeomorphism",
"Definition:Quotient Mapping (Topology)",
"Composite of Quotient Mappings in Topology is Quotient Mapping",
"Definition:Quotient Mapping (Topology)",
"... |
proofwiki-19970 | Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 2 | If $S_2$ is a Hausdorff space, then $S_1 / \RR_g$ is a Hausdorff space. | Continuous Surjection Induces Continuous Bijection from Quotient Space shows there exists a continuous bijection $f: S_1 / \RR_g \to S_2$.
Let $\eqclass {s_1}{\RR_g}, \eqclass {s_2}{\RR_g} \in S_1 / \RR_g$ such that $\eqclass {s_1}{\RR_g} \ne \eqclass {s_2}{\RR_g}$.
As $f$ is injective, $\map f { \eqclass {s_1}{\RR_g} ... | If $S_2$ is a [[Definition:Hausdorff Space|Hausdorff space]], then $S_1 / \RR_g$ is a [[Definition:Hausdorff Space|Hausdorff space]]. | [[Continuous Surjection Induces Continuous Bijection from Quotient Space]] shows there exists a [[Definition:Continuous Mapping|continuous]] [[Definition:Bijection|bijection]] $f: S_1 / \RR_g \to S_2$.
Let $\eqclass {s_1}{\RR_g}, \eqclass {s_2}{\RR_g} \in S_1 / \RR_g$ such that $\eqclass {s_1}{\RR_g} \ne \eqclass {s_2... | Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 2 | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space/Corollary_2 | https://proofwiki.org/wiki/Continuous_Surjection_Induces_Continuous_Bijection_from_Quotient_Space/Corollary_2 | [
"Continuous Surjection Induces Continuous Bijection from Quotient Space",
"Quotient Spaces (Topology)",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Continuous Surjection Induces Continuous Bijection from Quotient Space",
"Definition:Continuous Mapping",
"Definition:Bijection",
"Definition:Injection",
"Definition:T2 Space",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:Continuous Mapping",
... |
proofwiki-19971 | Linear Transformation is Fredholm Operator iff Pseudoinverse exists | Let $U, V$ be vector spaces.
Let $T: U \to V$ be a linear transformation.
Then $T$ is a Fredholm operator {{iff}} $T$ has a pseudoinverse. | Recall the definitions:
$S$ and $T$ are '''pseudoinverse''' to each other {{iff}}:
:$T \circ S - I_U$ is degenerate
and:
:$S \circ T - I_V$ is degenerate.
$T$ is a '''Fredholm operator''' {{iff}}:
:$(1): \quad \map \ker T$ is finite-dimensional
:$(2): \quad$ the quotient space $V / \Img T$ is finite-dimensional. | Let $U, V$ be [[Definition:Vector Space|vector spaces]].
Let $T: U \to V$ be a [[Definition:Linear Transformation on Vector Space|linear transformation]].
Then $T$ is a [[Definition:Fredholm Operator|Fredholm operator]] {{iff}} $T$ has a [[Definition:Pseudoinverse of Linear Transformation|pseudoinverse]]. | Recall the definitions:
$S$ and $T$ are '''[[Definition:Pseudoinverse of Linear Transformation|pseudoinverse]]''' to each other {{iff}}:
:$T \circ S - I_U$ is [[Definition:Degenerate Linear Transformation|degenerate]]
and:
:$S \circ T - I_V$ is [[Definition:Degenerate Linear Transformation|degenerate]].
$T$ is a '''... | Linear Transformation is Fredholm Operator iff Pseudoinverse exists | https://proofwiki.org/wiki/Linear_Transformation_is_Fredholm_Operator_iff_Pseudoinverse_exists | https://proofwiki.org/wiki/Linear_Transformation_is_Fredholm_Operator_iff_Pseudoinverse_exists | [
"Fredholm Operators",
"Linear Algebra",
"Functional Analysis"
] | [
"Definition:Vector Space",
"Definition:Linear Transformation/Vector Space",
"Definition:Fredholm Operator",
"Definition:Pseudoinverse of Linear Transformation"
] | [
"Definition:Pseudoinverse of Linear Transformation",
"Definition:Degenerate Linear Transformation",
"Definition:Degenerate Linear Transformation",
"Definition:Fredholm Operator",
"Definition:Dimension of Vector Space/Finite",
"Definition:Quotient Vector Space",
"Definition:Dimension of Vector Space/Fini... |
proofwiki-19972 | Simple Loop in Hausdorff Space is Homeomorphic to Quotient Space of Interval | Let $\struct {X, \tau_X }$ be a Hausdorff space.
Let $\gamma : \closedint 0 1 \to X$ be a simple loop.
Let $\sim$ be an equivalence relation on $\closedint 0 1$ defined by:
{{begin-eqn}}
{{eqn | o =
| r = \forall t_1 \in \openint 0 1 , t_2 \in \closedint 0 1 :
| rr = t_1 \sim t_2 \iff t_2 = t_1
}}
{{eqn | o... | Define $\tilde \gamma: \closedint 0 1 \to \Img \gamma$ as the restriction of $\gamma$ to $\closedint 0 1 \times \Img \gamma$.
Restriction of Mapping to Image is Surjection shows that $\tilde \gamma$ is surjective.
Subspace of Hausdorff Space is Hausdorff shows that $\struct {\Img \gamma, \tau_\gamma}$ is a Hausdorff sp... | Let $\struct {X, \tau_X }$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $\gamma : \closedint 0 1 \to X$ be a [[Definition:Simple Loop (Topology)|simple loop]].
Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on $\closedint 0 1$ defined by:
{{begin-eqn}}
{{eqn | o =
| r = \f... | Define $\tilde \gamma: \closedint 0 1 \to \Img \gamma$ as the [[Definition:Restriction of Mapping|restriction]] of $\gamma$ to $\closedint 0 1 \times \Img \gamma$.
[[Restriction of Mapping to Image is Surjection]] shows that $\tilde \gamma$ is [[Definition:Surjection|surjective]].
[[Subspace of Hausdorff Space is Hau... | Simple Loop in Hausdorff Space is Homeomorphic to Quotient Space of Interval | https://proofwiki.org/wiki/Simple_Loop_in_Hausdorff_Space_is_Homeomorphic_to_Quotient_Space_of_Interval | https://proofwiki.org/wiki/Simple_Loop_in_Hausdorff_Space_is_Homeomorphic_to_Quotient_Space_of_Interval | [
"Quotient Spaces (Topology)",
"Loops (Topology)"
] | [
"Definition:T2 Space",
"Definition:Loop (Topology)/Simple",
"Definition:Equivalence Relation",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Quotient Topology/Quotient Space",
"Definition:Topological Subspace",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition... | [
"Definition:Restriction/Mapping",
"Restriction of Mapping to Image is Surjection",
"Definition:Surjection",
"T2 Property is Hereditary",
"Definition:T2 Space",
"Closed Real Interval is Compact Space",
"Definition:Compact Space/Real Analysis",
"Continuous Mapping from Compact Space to Hausdorff Space i... |
proofwiki-19973 | Parameterization of Unit Circle is Simple Loop | Let $\mathbb S^1$ denote the unit circle whose center is at the origin of the Euclidean space $\R^2$.
Let $p: \closedint 0 1 \to \R^2$ be defined by:
:$\forall t \in \closedint 0 1 : \map p t = \tuple {\map \cos {2 \pi t}, \map \sin {2 \pi t} }$
Then $p$ is a simple loop with image equal to $\mathbb S^1$. | Parametric Equation of Circle shows that for all $r \in \R$, the point $\tuple {\map \cos r, \map \sin r}$ lies on the unit circle $\mathbb S^1$.
Parametric Equation of Circle also shows that for all points $\tuple {x, y}$ on the unit circle $\mathbb S^1$, the point can be expressed as $\tuple {x, y} = \tuple {\map \co... | Let $\mathbb S^1$ denote the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Real Euclidean Space|Euclidean space]] $\R^2$.
Let $p: \closedint 0 1 \to \R^2$ be defined by:
:$\forall t \in \closedint 0 1 : \map p t = \tuple ... | [[Equation of Circle/Parametric|Parametric Equation of Circle]] shows that for all $r \in \R$, the [[Definition:Point|point]] $\tuple {\map \cos r, \map \sin r}$ lies on the [[Definition:Unit Circle|unit circle]] $\mathbb S^1$.
[[Equation of Circle/Parametric|Parametric Equation of Circle]] also shows that for all [[D... | Parameterization of Unit Circle is Simple Loop | https://proofwiki.org/wiki/Parameterization_of_Unit_Circle_is_Simple_Loop | https://proofwiki.org/wiki/Parameterization_of_Unit_Circle_is_Simple_Loop | [
"Circles",
"Loops (Topology)"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Euclidean Space/Real",
"Definition:Loop (Topology)/Simple",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Equation of Circle/Parametric",
"Definition:Point",
"Definition:Unit Circle",
"Equation of Circle/Parametric",
"Definition:Point",
"Definition:Unit Circle",
"Definition:Point",
"Sine and Cosine are Periodic on Reals",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definiti... |
proofwiki-19974 | Countably Additive Function Dichotomy by Empty Set | Let $\AA$ be a $\sigma$-algebra.
Let $\overline \R$ denote the extended set of real numbers.
Let $f: \AA \to \overline \R$ be a function be a countably additive function.
Then the exactly one of the following is true:
:$\paren 1$: $\map f \O = 0$
:$\paren 2$: $\map f \O = + \infty$. Moreover, $\map f A = + \infty$ for ... | Suppose $\map f \O \ne 0$.
Then:
{{begin-eqn}}
{{eqn | l = \map f \O
| r = \map f {\bigcup_{n \mathop \in \N} \O}
}}
{{eqn | r = \sum_{n \mathop \in \N} \map f \O
| c = {{Defof|Countably Additive Function}}
}}
{{eqn | o = \in
| r = \set {+\infty, -\infty}
}}
{{end-eqn}}
{{qed|lemma}}
Furthermore, for ... | Let $\AA$ be a [[Definition:Sigma-Algebra|$\sigma$-algebra]].
Let $\overline \R$ denote the [[Definition:Extended Real Number Line|extended set of real numbers]].
Let $f: \AA \to \overline \R$ be a [[Definition:Mapping|function]] be a [[Definition:Countably Additive Function|countably additive function]].
Then the ... | Suppose $\map f \O \ne 0$.
Then:
{{begin-eqn}}
{{eqn | l = \map f \O
| r = \map f {\bigcup_{n \mathop \in \N} \O}
}}
{{eqn | r = \sum_{n \mathop \in \N} \map f \O
| c = {{Defof|Countably Additive Function}}
}}
{{eqn | o = \in
| r = \set {+\infty, -\infty}
}}
{{end-eqn}}
{{qed|lemma}}
Furthermore, ... | Countably Additive Function Dichotomy by Empty Set | https://proofwiki.org/wiki/Countably_Additive_Function_Dichotomy_by_Empty_Set | https://proofwiki.org/wiki/Countably_Additive_Function_Dichotomy_by_Empty_Set | [
"Countably Additive Functions",
"Empty Set"
] | [
"Definition:Sigma-Algebra",
"Definition:Extended Real Number Line",
"Definition:Mapping",
"Definition:Countably Additive Function"
] | [
"Category:Countably Additive Functions",
"Category:Empty Set"
] |
proofwiki-19975 | Simple Loop Image Equals Set Homeomorphic to Circle | Let $\struct { X, \tau_X }$ be a Hausdorff space.
Let $C \subseteq X$ be a subset of $X$.
Let $\tau_C$ be the subspace topology on $C$ induced by $\tau_X$.
Let $\mathbb S^1$ denote the unit circle whose center is at the origin of the Euclidean space $\R^2$.
Let $\tau_{\mathbb S^1}$ be the subspace topology on $\mathbb ... | === Sufficient condition ===
Let $\sim$ be the equivalence relation on the closed real interval $\closedint 0 1$ defined by:
{{begin-eqn}}
{{eqn | q = \forall t_1 \in \openint 0 1, t_2 \in \closedint 0 1
| l = t_1 \sim t_2
| o = \iff
| r = t_2 = t_1
}}
{{eqn | q = \forall t_1 \in \set {0, 1}, t_2 \in ... | Let $\struct { X, \tau_X }$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $C \subseteq X$ be a [[Definition:Subset|subset]] of $X$.
Let $\tau_C$ be the [[Definition:Subspace Topology|subspace topology]] on $C$ induced by $\tau_X$.
Let $\mathbb S^1$ denote the [[Definition:Unit Circle|unit circle]] whose [... | === Sufficient condition ===
Let $\sim$ be the [[Definition:Equivalence Relation|equivalence relation]] on the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 1$ defined by:
{{begin-eqn}}
{{eqn | q = \forall t_1 \in \openint 0 1, t_2 \in \closedint 0 1
| l = t_1 \sim t_2
| o = \iff
... | Simple Loop Image Equals Set Homeomorphic to Circle | https://proofwiki.org/wiki/Simple_Loop_Image_Equals_Set_Homeomorphic_to_Circle | https://proofwiki.org/wiki/Simple_Loop_Image_Equals_Set_Homeomorphic_to_Circle | [
"Loops (Topology)",
"Circles",
"Examples of Homeomorphisms"
] | [
"Definition:T2 Space",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Euclidean Space/Real",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology/Real Number... | [
"Definition:Equivalence Relation",
"Definition:Real Interval/Closed",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Quotient Topology/Quotient Space",
"Definition:Quotient Topology",
"Definition:Identification Topology",
"Definition:Identification Topology/Identification M... |
proofwiki-19976 | GCD of Sum and Difference of Coprime Integers | Let $a, b \in \Z$ be coprime integers.
Then:
:$\gcd \set {a + b, a - b} = 1 \text { or } 2$
where:
:$\gcd$ denotes greatest common divisor. | Let:
:$d = \gcd \set {a + b, a - b}$
We have:
{{begin-eqn}}
{{eqn | l = \gcd \set {a + b, a - b}
| r = \gcd \set {a + b, a + b - 2 b}
| c =
}}
{{eqn | o = \divides
| r = 2 b
| c = GCD of Integer with Integer + n
}}
{{eqn | l = \gcd \set {a + b, a - b}
| r = \gcd \set {-\paren {a + b}, a -... | Let $a, b \in \Z$ be [[Definition:Coprime Integers|coprime integers]].
Then:
:$\gcd \set {a + b, a - b} = 1 \text { or } 2$
where:
:$\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]. | Let:
:$d = \gcd \set {a + b, a - b}$
We have:
{{begin-eqn}}
{{eqn | l = \gcd \set {a + b, a - b}
| r = \gcd \set {a + b, a + b - 2 b}
| c =
}}
{{eqn | o = \divides
| r = 2 b
| c = [[GCD of Integer with Integer + n]]
}}
{{eqn | l = \gcd \set {a + b, a - b}
| r = \gcd \set {-\paren {a + b... | GCD of Sum and Difference of Coprime Integers | https://proofwiki.org/wiki/GCD_of_Sum_and_Difference_of_Coprime_Integers | https://proofwiki.org/wiki/GCD_of_Sum_and_Difference_of_Coprime_Integers | [
"Greatest Common Divisor",
"Coprime Integers"
] | [
"Definition:Coprime/Integers",
"Definition:Greatest Common Divisor/Integers"
] | [
"GCD of Integer with Integer + n",
"GCD for Negative Integers",
"GCD of Integer with Integer + n",
"GCD of Integers with Common Divisor"
] |
proofwiki-19977 | Jordan Curve Image Equals Set Homeomorphic to Circle | Let $C \subseteq \R^2$ be a subset of the Euclidean space $\R^2$.
Let $\mathbb S^1$ denote the unit circle whose center is at the origin of the Euclidean space $\R^2$.
Let $\tau_C$ and $\tau_{\mathbb S^1}$ be the subspace topologies on $C$ respectively $\mathbb S^1$ induced by the Euclidean topology on $\R^2$.
Then $\s... | By definition of simple loop, a Jordan curve is a simple loop in $\R^2$.
Euclidean Space is Complete Metric Space and Metric Space is Hausdorff shows that $\R^2$ is a Hausdorff space.
The result now follows from Simple Loop Image Equals Set Homeomorphic to Circle.
{{qed}}
Category:Circles
Category:Examples of Homeomorp... | Let $C \subseteq \R^2$ be a [[Definition:Subset|subset]] of the [[Definition:Real Euclidean Space|Euclidean space]] $\R^2$.
Let $\mathbb S^1$ denote the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Real Euclidean Space|Eu... | By definition of [[Definition:Simple Loop (Topology)|simple loop]], a [[Definition:Jordan Curve|Jordan curve]] is a [[Definition:Simple Loop (Topology)|simple loop]] in $\R^2$.
[[Euclidean Space is Complete Metric Space]] and [[Metric Space is Hausdorff]] shows that $\R^2$ is a [[Definition:Hausdorff Space|Hausdorff s... | Jordan Curve Image Equals Set Homeomorphic to Circle | https://proofwiki.org/wiki/Jordan_Curve_Image_Equals_Set_Homeomorphic_to_Circle | https://proofwiki.org/wiki/Jordan_Curve_Image_Equals_Set_Homeomorphic_to_Circle | [
"Circles",
"Examples of Homeomorphisms",
"Jordan Curves"
] | [
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Euclidean Space/Real",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Ho... | [
"Definition:Loop (Topology)/Simple",
"Definition:Jordan Curve",
"Definition:Loop (Topology)/Simple",
"Euclidean Space is Complete Metric Space",
"Metric Space is T2",
"Definition:T2 Space",
"Simple Loop Image Equals Set Homeomorphic to Circle",
"Category:Circles",
"Category:Examples of Homeomorphism... |
proofwiki-19978 | Birkhoff's Ergodic Theorem | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Let $f: X \to \overline \R$ be a $\mu$-integrable function.
Then a $\mu$-integrable function $\tilde f$ exists such that:
:$\forall x \in X : \map {\tilde f} {T x} = \map {\tilde f} x$
and:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 n \sum_{n \mat... | Let $f_0 : X \to \R$ be a bounded measurable function.
For $N \in \N_{>0}$, let:
:$\ds \map {A_N} {f_0} = \frac 1 N \sum_{n \mathop = 0}^{N-1} f_0 \circ T^n$
By $L^1$ Mean Ergodic Theorem, there is a $\mu$-integrable function $F_0$ such that:
:$F_0 \circ T = F_0$
and:
:$\ds \lim_{N \mathop \to \infty}\norm {F_0 - \map ... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Let $f: X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]].
Then a [[Definition:Measure-Integrable Function|$\mu$-integrable function]] $\tilde f$ exi... | Let $f_0 : X \to \R$ be a [[Definition:Bounded Mapping|bounded]] [[Definition:Measurable Function|measurable function]].
For $N \in \N_{>0}$, let:
:$\ds \map {A_N} {f_0} = \frac 1 N \sum_{n \mathop = 0}^{N-1} f_0 \circ T^n$
By [[L1 Mean Ergodic Theorem|$L^1$ Mean Ergodic Theorem]], there is a [[Definition:Measure-Int... | Birkhoff's Ergodic Theorem | https://proofwiki.org/wiki/Birkhoff's_Ergodic_Theorem | https://proofwiki.org/wiki/Birkhoff's_Ergodic_Theorem | [
"Dynamical Systems Theory",
"Ergodic Theory"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Convergence Almost Everywhere",
"Definition:Lp Norm",
"Definition:Conditional Expectation/General Case/Sigma-Algebra"
] | [
"Definition:Bounded Mapping",
"Definition:Measurable Function",
"L1 Mean Ergodic Theorem",
"Definition:Integrable Function/Measure Space",
"Maximal Ergodic Theorem",
"Definition:Integrable Function/Measure Space",
"L1 Mean Ergodic Theorem",
"Definition:Integrable Function/Measure Space",
"Definition... |
proofwiki-19979 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11 | :$\forall n \in \N : \dfrac {a - \paren {p^{n + 1} - 1} b } {p^{n + 1} } \le r_n \le \dfrac a {p^{n + 1} }$ | We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| m = \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b}
| mo= \le
| r = p^{n + 1} - 1
}}
{{eqn | ll= \leadsto
| l = -\paren {p^{n + 1} - 1}
| o = \le
| m = \paren{p^{n + 1} \dfrac {r_n} b} - \dfrac a b
| mo= \le
| r = 0
}}
... | :$\forall n \in \N : \dfrac {a - \paren {p^{n + 1} - 1} b } {p^{n + 1} } \le r_n \le \dfrac a {p^{n + 1} }$ | We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| m = \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b}
| mo= \le
| r = p^{n + 1} - 1
}}
{{eqn | ll= \leadsto
| l = -\paren {p^{n + 1} - 1}
| o = \le
| m = \paren{p^{n + 1} \dfrac {r_n} b} - \dfrac a b
| mo= \le
| r = 0
}}
... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_11 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_11 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Category:Canonical P-adic Expansion of Rational is Eventually Periodic"
] |
proofwiki-19980 | Jordan-Schönflies Theorem | Let $\gamma : \closedint 0 1 \to \R^2$ be a Jordan curve.
Let $\Img \gamma$ denote the image of $\gamma$, $\Int \gamma$ denote the interior of $\gamma$, and $\Ext \gamma$ denote the exterior of $\gamma$.
Let $\mathbb S^1$ denote the unit circle whose center is at the origin $\mathbf 0$ of the Euclidean space $\R^2$..
L... | {{proof wanted}}
{{Namedfor|Marie Ennemond Camille Jordan|name2 = Arthur Moritz Schönflies|cat = Jordan, C|cat2 = Schönflies}} | Let $\gamma : \closedint 0 1 \to \R^2$ be a [[Definition:Jordan Curve|Jordan curve]].
Let $\Img \gamma$ denote the [[Definition:Image of Mapping|image]] of $\gamma$, $\Int \gamma$ denote the [[Definition:Interior of Jordan Curve|interior]] of $\gamma$, and $\Ext \gamma$ denote the [[Definition:Exterior of Jordan Curve... | {{proof wanted}}
{{Namedfor|Marie Ennemond Camille Jordan|name2 = Arthur Moritz Schönflies|cat = Jordan, C|cat2 = Schönflies}} | Jordan-Schönflies Theorem | https://proofwiki.org/wiki/Jordan-Schönflies_Theorem | https://proofwiki.org/wiki/Jordan-Schönflies_Theorem | [
"Jordan Curves"
] | [
"Definition:Jordan Curve",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Jordan Curve/Interior",
"Definition:Jordan Curve/Exterior",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Euclidean Space/Real",
"Definition:Open Ball/No... | [] |
proofwiki-19981 | Zeroes of Sine and Cosine/Cosine | :$\cos x = 0$ {{iff}} $x = \paren {n + \dfrac 1 2} \pi$ for some $n \in \Z$. | From Sine and Cosine are Periodic on Reals: Corollary:
$\cos x$ is:
:strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$
and:
:strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
The result follows directly from Sine and Cosine are Periodic on Reals.
{{qed}}
Category:Z... | :$\cos x = 0$ {{iff}} $x = \paren {n + \dfrac 1 2} \pi$ for some $n \in \Z$. | From [[Sine and Cosine are Periodic on Reals/Corollary|Sine and Cosine are Periodic on Reals: Corollary]]:
$\cos x$ is:
:[[Definition:Strictly Positive|strictly positive]] on the [[Definition:Open Real Interval|interval]] $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$
and:
:[[Definition:Strictly Negative|strictly negative]... | Zeroes of Sine and Cosine/Cosine | https://proofwiki.org/wiki/Zeroes_of_Sine_and_Cosine/Cosine | https://proofwiki.org/wiki/Zeroes_of_Sine_and_Cosine/Cosine | [
"Zeroes of Sine and Cosine"
] | [] | [
"Sine and Cosine are Periodic on Reals/Corollary",
"Definition:Strictly Positive",
"Definition:Real Interval/Open",
"Definition:Strictly Negative",
"Definition:Real Interval/Open",
"Sine and Cosine are Periodic on Reals",
"Category:Zeroes of Sine and Cosine"
] |
proofwiki-19982 | Zeroes of Sine and Cosine/Sine | :$\sin x = 0$, {{iff}} $x = n \pi$ for some $n \in \Z$. | From Sine and Cosine are Periodic on Reals: Corollary:
$\sin x$ is:
:strictly positive on the interval $\openint 0 \pi$
and:
:strictly negative on the interval $\openint \pi {2 \pi}$
The result follows directly from Sine and Cosine are Periodic on Reals.
{{qed}}
Category:Zeroes of Sine and Cosine
1zgall37peqab11t4tjgjd... | :$\sin x = 0$, {{iff}} $x = n \pi$ for some $n \in \Z$. | From [[Sine and Cosine are Periodic on Reals/Corollary|Sine and Cosine are Periodic on Reals: Corollary]]:
$\sin x$ is:
:[[Definition:Strictly Positive|strictly positive]] on the [[Definition:Open Real Interval|interval]] $\openint 0 \pi$
and:
:[[Definition:Strictly Negative|strictly negative]] on the [[Definition:Ope... | Zeroes of Sine and Cosine/Sine | https://proofwiki.org/wiki/Zeroes_of_Sine_and_Cosine/Sine | https://proofwiki.org/wiki/Zeroes_of_Sine_and_Cosine/Sine | [
"Zeroes of Sine and Cosine"
] | [] | [
"Sine and Cosine are Periodic on Reals/Corollary",
"Definition:Strictly Positive",
"Definition:Real Interval/Open",
"Definition:Strictly Negative",
"Definition:Real Interval/Open",
"Sine and Cosine are Periodic on Reals",
"Category:Zeroes of Sine and Cosine"
] |
proofwiki-19983 | Sine and Cosine are Periodic on Reals/Pi | The real number $\pi$ (called '''pi''', pronounced '''pie''') is uniquely defined as:
:$\pi := \dfrac p 2$
where $p \in \R$ is the period of $\sin$ and $\cos$. | From the Real Cosine Function is Periodic and Real Sine Function is Periodic, we have that $\cos x$ and $\sin x$ are periodic on $\R$ with the same period.
If we denote the period of $\cos x$ and $\sin x$ as $p$, it follows that $\pi = \dfrac p 2$ is uniquely defined.
{{qed}} | The [[Definition:Real Number|real number]] [[Definition:Pi|$\pi$ (called '''pi''', pronounced '''pie''')]] is uniquely defined as:
:$\pi := \dfrac p 2$
where $p \in \R$ is the [[Definition:Period of Periodic Real Function|period]] of $\sin$ and $\cos$. | From the [[Real Cosine Function is Periodic]] and [[Real Sine Function is Periodic]], we have that $\cos x$ and $\sin x$ are [[Definition:Periodic Real Function|periodic]] on $\R$ with the same [[Definition:Period of Periodic Real Function|period]].
If we denote the [[Definition:Period of Periodic Real Function|period... | Sine and Cosine are Periodic on Reals/Pi/Proof 1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi/Proof_1 | [
"Sine and Cosine are Periodic on Reals"
] | [
"Definition:Real Number",
"Definition:Pi",
"Definition:Periodic Real Function/Period"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Sine and Cosine are Periodic on Reals/Sine",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Real Function/Period"
] |
proofwiki-19984 | Sine and Cosine are Periodic on Reals/Pi | The real number $\pi$ (called '''pi''', pronounced '''pie''') is uniquely defined as:
:$\pi := \dfrac p 2$
where $p \in \R$ is the period of $\sin$ and $\cos$. | By Cosine of Zero is One:
:$\cos 0 = 1$
By Cosine of 2 is Strictly Negative:
:$\cos 2 < 0$
Thus by {{Corollary|Intermediate Value Theorem}} there exists an $h \in \openint 0 2$ such that:
:$\cos h = 0$
By Sine of Sum for all $x \in \R$:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \map \sin {x - h} \cos h + \map \cos {... | The [[Definition:Real Number|real number]] [[Definition:Pi|$\pi$ (called '''pi''', pronounced '''pie''')]] is uniquely defined as:
:$\pi := \dfrac p 2$
where $p \in \R$ is the [[Definition:Period of Periodic Real Function|period]] of $\sin$ and $\cos$. | By [[Cosine of Zero is One]]:
:$\cos 0 = 1$
By [[Cosine of 2 is Strictly Negative]]:
:$\cos 2 < 0$
Thus by {{Corollary|Intermediate Value Theorem}} there exists an $h \in \openint 0 2$ such that:
:$\cos h = 0$
By [[Sine of Sum]] for all $x \in \R$:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \map \sin {x - h} \co... | Sine and Cosine are Periodic on Reals/Pi/Proof 2 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi/Proof_2 | [
"Sine and Cosine are Periodic on Reals"
] | [
"Definition:Real Number",
"Definition:Pi",
"Definition:Periodic Real Function/Period"
] | [
"Cosine of Zero is One",
"Cosine of 2 is Strictly Negative",
"Sine of Sum",
"Cosine of Sum",
"Sum of Squares of Sine and Cosine",
"Definition:Periodic Function/Real",
"Nonconstant Periodic Function with no Period is Discontinuous Everywhere",
"Definition:Periodic Real Function/Period",
"Definition:P... |
proofwiki-19985 | Sine and Cosine are Periodic on Reals/Sine | The real sine function is periodic
with the same period as the real cosine function. | Since Real Cosine Function is Periodic, let $K$ be its period.
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because Cosine of Zero is One:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | l = \sin^2 K
| r = 0
| c ... | The [[Definition:Real Sine Function|real sine function]] is [[Definition:Periodic Real Function|periodic]]
with the same [[Definition:Period of Periodic Real Function|period]] as the [[Definition:Real Cosine Function|real cosine function]]. | Since [[Real Cosine Function is Periodic]], let $K$ be its [[Definition:Period of Periodic Real Function|period]].
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because [[Cosine of Zero is One]]:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = [[Sum of Squares of Si... | Sine and Cosine are Periodic on Reals/Sine/Proof 1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_1 | [
"Sine and Cosine are Periodic on Reals"
] | [
"Definition:Sine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Cosine/Real Function"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Periodic Real Function/Period",
"Cosine of Zero is One",
"Sum of Squares of Sine and Cosine",
"Sine of Sum",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Real Function/Period",
"Sine ... |
proofwiki-19986 | Sine and Cosine are Periodic on Reals/Sine | The real sine function is periodic
with the same period as the real cosine function. | Since Real Cosine Function is Periodic, let $L$ be its period.
From Primitive of Cosine Function:
{{:Primitive of Cosine Function}}
for any constant $C$.
Therefore $\sin x$ is a Primitive of $\cos x$, for the special case of $C = 0$.
From Primitive of Periodic Real Function, it follows that $\sin x$ is periodic with pe... | The [[Definition:Real Sine Function|real sine function]] is [[Definition:Periodic Real Function|periodic]]
with the same [[Definition:Period of Periodic Real Function|period]] as the [[Definition:Real Cosine Function|real cosine function]]. | Since [[Real Cosine Function is Periodic]], let $L$ be its [[Definition:Period of Periodic Real Function|period]].
From [[Primitive of Cosine Function]]:
{{:Primitive of Cosine Function}}
for any constant $C$.
Therefore $\sin x$ is a [[Definition:Primitive (Calculus)|Primitive]] of $\cos x$, for the special case of $... | Sine and Cosine are Periodic on Reals/Sine/Proof 2 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_2 | [
"Sine and Cosine are Periodic on Reals"
] | [
"Definition:Sine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Cosine/Real Function"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Periodic Real Function/Period",
"Primitive of Cosine Function",
"Definition:Primitive (Calculus)",
"Primitive of Periodic Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] |
proofwiki-19987 | Sine and Cosine are Periodic on Reals/Cosine | :$\exists L \in \R_{\neq 0}: \forall x \in \R: \cos x = \map \cos {x + L}$ | From Real Cosine Function has Zeroes, the cosine function has at least one positive zero.
Therefore there exists a Greatest Lower Bound $\eta \in \R_{>0}$ to the set of positive zeroes.
Since Cosine Function is Continuous, $\eta$ is a zero.
Because Cosine Function is Even:
:$\cos \eta = \map \cos {-\eta} = 0$
By defini... | :$\exists L \in \R_{\neq 0}: \forall x \in \R: \cos x = \map \cos {x + L}$ | From [[Real Cosine Function has Zeroes]], the [[Definition:Real Cosine Function|cosine]] function has at least one [[Definition:Strictly Positive Real Number|positive]] [[Definition:Zero of Function|zero]].
Therefore there exists a [[Definition:Greatest Lower Bound|Greatest Lower Bound]] $\eta \in \R_{>0}$ to the [[De... | Sine and Cosine are Periodic on Reals/Cosine | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Cosine | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Cosine | [
"Sine and Cosine are Periodic on Reals"
] | [] | [
"Real Cosine Function has Zeroes",
"Definition:Cosine/Real Function",
"Definition:Strictly Positive/Real Number",
"Definition:Root of Mapping",
"Definition:Infimum of Set/Real Numbers",
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Root of Mapping",
"Cosine Function is Co... |
proofwiki-19988 | P-Sequence Space admits Schauder Basis | Let $1 \le p < \infty$.
Let $\ell^p$ be the $p$-sequence space.
Let $\sequence {\mathbf e_n}_{n \mathop \in \N } \in \ell^p$ be a sequence such that:
:$\mathbf e_n = \tuple {\underbrace{0, \ldots, 0}_n, 1, 0, \ldots}$
Then $\set {\mathbf e_n : n \in \N}$ is a Schauder basis for $\ell^p$. | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} = \tuple {x_1, x_2, x_3, \ldots} \in \ell^p$. | Let $1 \le p < \infty$.
Let $\ell^p$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $\sequence {\mathbf e_n}_{n \mathop \in \N } \in \ell^p$ be a [[Definition:Sequence|sequence]] such that:
:$\mathbf e_n = \tuple {\underbrace{0, \ldots, 0}_n, 1, 0, \ldots}$
Then $\set {\mathbf e_n : n \in \N}$ is a... | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} = \tuple {x_1, x_2, x_3, \ldots} \in \ell^p$. | P-Sequence Space admits Schauder Basis | https://proofwiki.org/wiki/P-Sequence_Space_admits_Schauder_Basis | https://proofwiki.org/wiki/P-Sequence_Space_admits_Schauder_Basis | [
"Schauder Bases",
"P-Sequence Spaces",
"Bases of Vector Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Sequence",
"Definition:Schauder Basis"
] | [] |
proofwiki-19989 | Vector Addition on Normed Vector Space is Continuous | Let $\struct {X, \norm {\, \cdot \,}_X }$ be a normed vector space.
Let $\struct {X \times X, \norm {\, \cdot \,}_P }$ be the direct product of $X$ and $X$ with the direct product norm $\norm {\, \cdot \,}_P$.
Let $+_{\scriptscriptstyle X} : X \times X \to X$ be the vector addition defined on $X$.
Then $+_{\scriptscri... | Let $x_0, y_0 \in X$.
Let $\epsilon \in \R_{>0}$.
For $a, b \in X$, let $a-_{\scriptscriptstyle X} b$ denote the sum $a +_{\scriptscriptstyle X} \paren { -b }$, where $-b$ is the inverse vector of $b$ in $X$.
To show that $+_{\scriptscriptstyle X}$ is continuous, let $x, y \in X$ such that $\norm { x_0 -_{\scriptscript... | Let $\struct {X, \norm {\, \cdot \,}_X }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X \times X, \norm {\, \cdot \,}_P }$ be the [[Definition:Direct Product of Vector Spaces/Finite Case|direct product]] of $X$ and $X$ with the [[Definition:Direct Product Norm|direct product norm]] $\nor... | Let $x_0, y_0 \in X$.
Let $\epsilon \in \R_{>0}$.
For $a, b \in X$, let $a-_{\scriptscriptstyle X} b$ denote the sum $a +_{\scriptscriptstyle X} \paren { -b }$, where $-b$ is the [[Definition:Inverse Element|inverse vector]] of $b$ in $X$.
To show that $+_{\scriptscriptstyle X}$ is [[Definition:Continuous Mapping (N... | Vector Addition on Normed Vector Space is Continuous | https://proofwiki.org/wiki/Vector_Addition_on_Normed_Vector_Space_is_Continuous | https://proofwiki.org/wiki/Vector_Addition_on_Normed_Vector_Space_is_Continuous | [
"Continuous Mappings on Normed Vector Spaces",
"Vector Addition"
] | [
"Definition:Normed Vector Space",
"Definition:Direct Product of Vector Spaces/Finite Case",
"Definition:Direct Product Norm",
"Definition:Vector Addition/Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)"
] | [
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Direct Product Norm",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] |
proofwiki-19990 | Abel's Summation Formula | Let $\sequence {a_n}_{n \in \N_{>0} }$ be a sequence in $\R$.
Let $f : \R_{\ge 1} \to \R$ be a continuously differentiable function.
Let $A : \R_{\ge 1} \to \R$ be defined as:
:$\ds \map A x := \sum_{1 \mathop \le n \mathop \le x} a_n$
Then for all $x \in \R_{\ge 1}$:
:$\ds \sum_{1 \mathop \le n \mathop \le x} a_n \map... | {{ProofWanted}}
{{Namedfor|Niels Henrik Abel|cat = Abel}}
Category:Integral Calculus
Category:Analytic Number Theory
s5ovd2lpotvbn0usbfr1opzq5rli6m8 | Let $\sequence {a_n}_{n \in \N_{>0} }$ be a [[Definition:Sequence|sequence]] in $\R$.
Let $f : \R_{\ge 1} \to \R$ be a [[Definition:Continuously Differentiable Real-Valued Function|continuously differentiable function]].
Let $A : \R_{\ge 1} \to \R$ be defined as:
:$\ds \map A x := \sum_{1 \mathop \le n \mathop \le x}... | {{ProofWanted}}
{{Namedfor|Niels Henrik Abel|cat = Abel}}
[[Category:Integral Calculus]]
[[Category:Analytic Number Theory]]
s5ovd2lpotvbn0usbfr1opzq5rli6m8 | Abel's Summation Formula | https://proofwiki.org/wiki/Abel's_Summation_Formula | https://proofwiki.org/wiki/Abel's_Summation_Formula | [
"Integral Calculus",
"Analytic Number Theory"
] | [
"Definition:Sequence",
"Definition:Continuously Differentiable/Real-Valued Function"
] | [
"Category:Integral Calculus",
"Category:Analytic Number Theory"
] |
proofwiki-19991 | Babbage's Congruence | Let $p$ be a prime number.
Let $a, b \in \Z$ be integers.
Then:
:$\dbinom {a p} {b p} \equiv \dbinom a b \pmod {p^2}$
where $\dbinom a b$ denotes a binomial coefficient. | By Lucas' Theorem, for $a, b \in \mathbb Z_{\ge 0}$:
:$\dbinom {a p} {b p} \equiv \dbinom {\floor {a p / p}} {\floor {b p / p}} \dbinom {a \mod p} {b \mod p} \pmod p$
Proof by Mathematical Induction on $a$:
For all $a \in \Z$, let $\map P a$ be the proposition:
:$\forall b \in \Z: \dbinom {a p} {b p} \equiv \dbinom a b... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Then:
:$\dbinom {a p} {b p} \equiv \dbinom a b \pmod {p^2}$
where $\dbinom a b$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]. | By [[Lucas' Theorem]], for $a, b \in \mathbb Z_{\ge 0}$:
:$\dbinom {a p} {b p} \equiv \dbinom {\floor {a p / p}} {\floor {b p / p}} \dbinom {a \mod p} {b \mod p} \pmod p$
Proof by [[Proof by Mathematical Induction|Mathematical Induction]] on $a$:
For all $a \in \Z$, let $\map P a$ be the [[Definition:Proposition|prop... | Babbage's Congruence | https://proofwiki.org/wiki/Babbage's_Congruence | https://proofwiki.org/wiki/Babbage's_Congruence | [
"Binomial Coefficients"
] | [
"Definition:Prime Number",
"Definition:Integer",
"Definition:Binomial Coefficient"
] | [
"Lucas' Theorem",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Lucas' Theorem",
"Lucas' Theorem",
"Lucas' Theorem"
] |
proofwiki-19992 | Divisibility of Numerator of Sum of Sequence of Reciprocals | Let $p$ be a prime number such that $p > 3$.
Consider the sum of the finite sequence of reciprocals as follows:
:$S = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 {p - 1}$
Let $S$ be expressed as a fraction in canonical form, that is:
:$S = \dfrac a b$
where $a$ and $b$ are coprime.
Then:
:$p^2 \divides a$
where $\d... | === Lemma ===
{{:Divisibility of Numerator of Sum of Sequence of Reciprocals/Lemma}}
{{ProofWanted}} | Let $p$ be a [[Definition:Prime Number|prime number]] such that $p > 3$.
Consider the [[Definition:Rational Addition|sum]] of the [[Definition:Finite Sequence|finite sequence]] of [[Definition:Reciprocal|reciprocals]] as follows:
:$S = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 {p - 1}$
Let $S$ be expressed as ... | === [[Divisibility of Numerator of Sum of Sequence of Reciprocals/Lemma|Lemma]] ===
{{:Divisibility of Numerator of Sum of Sequence of Reciprocals/Lemma}}
{{ProofWanted}} | Divisibility of Numerator of Sum of Sequence of Reciprocals | https://proofwiki.org/wiki/Divisibility_of_Numerator_of_Sum_of_Sequence_of_Reciprocals | https://proofwiki.org/wiki/Divisibility_of_Numerator_of_Sum_of_Sequence_of_Reciprocals | [
"Divisibility of Numerator of Sum of Sequence of Reciprocals",
"Divisibility",
"Reciprocals"
] | [
"Definition:Prime Number",
"Definition:Addition/Rational Numbers",
"Definition:Finite Sequence",
"Definition:Reciprocal",
"Definition:Fraction",
"Definition:Rational Number/Canonical Form",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisibility of Numerator of Sum of Sequence of Reciprocals/Lemma"
] |
proofwiki-19993 | Integral Expression of Harmonic Number | Let $\sequence {H_n}_{n \mathop \in \N}$ be the harmonic numbers.
Then:
:$\ds H_n = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u$
where $\floor u$ denotes the floor of $u$. | Observe that:
:$\ds \forall x \in \R_{\ge 1} : \floor x = \sum_{1 \mathop \le k \mathop \le x} 1$
Let $f: \R_{\ge 1} \to \R$ be defined as:
:$\ds \map f x := \dfrac 1 x$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \frac 1 k
| r = \sum_{k \mathop = 1}^n 1 \cdot \map f k
}}
{{eqn | r = \floor n \map f n ... | Let $\sequence {H_n}_{n \mathop \in \N}$ be the [[Definition:Harmonic Numbers|harmonic numbers]].
Then:
:$\ds H_n = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u$
where $\floor u$ denotes the [[Definition:Floor Function|floor]] of $u$. | Observe that:
:$\ds \forall x \in \R_{\ge 1} : \floor x = \sum_{1 \mathop \le k \mathop \le x} 1$
Let $f: \R_{\ge 1} \to \R$ be defined as:
:$\ds \map f x := \dfrac 1 x$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \frac 1 k
| r = \sum_{k \mathop = 1}^n 1 \cdot \map f k
}}
{{eqn | r = \floor n \map f... | Integral Expression of Harmonic Number | https://proofwiki.org/wiki/Integral_Expression_of_Harmonic_Number | https://proofwiki.org/wiki/Integral_Expression_of_Harmonic_Number | [
"Harmonic Numbers",
"Analytic Number Theory"
] | [
"Definition:Harmonic Numbers",
"Definition:Floor Function"
] | [
"Abel's Summation Formula",
"Derivative of Reciprocal",
"Category:Harmonic Numbers",
"Category:Analytic Number Theory"
] |
proofwiki-19994 | Normed Vector Space is Hausdorff | Let $\struct {X, \norm {\, \cdot \,}_X }$ be a normed vector space.
Let $\tau$ be the topology on $X$ that consists of all open sets in $X$.
That is, $U \in \tau$ {{iff}}:
:$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$
where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.
Th... | From Metric Induced by Norm is Metric, it follows that the norm $\norm {\,\cdot\,}$ induces a metric $d$ on $X$.
From the definition of open set in normed vector space, it follows that $\tau$ is the topology induced by the metric $d$.
The result now follows from Metric Space is Hausdorff.
{{qed}}
Category:Normed Vector... | Let $\struct {X, \norm {\, \cdot \,}_X }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\tau$ be the [[Definition:Topology|topology]] on $X$ that consists of all [[Definition:Open Set in Normed Vector Space|open sets]] in $X$.
That is, $U \in \tau$ {{iff}}:
:$\forall x \in U: \exists \epsilon \in... | From [[Metric Induced by Norm is Metric]], it follows that the norm $\norm {\,\cdot\,}$ [[Definition:Metric Induced by Norm|induces a metric]] $d$ on $X$.
From the definition of [[Definition:Open Set in Normed Vector Space|open set in normed vector space]], it follows that $\tau$ is the [[Definition:Topology Induced b... | Normed Vector Space is Hausdorff | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff | [
"Normed Vector Spaces",
"Hausdorff Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Topology",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:T2 Space"
] | [
"Metric Induced by Norm is Metric",
"Definition:Metric Induced by Norm",
"Definition:Open Set/Normed Vector Space",
"Definition:Topology Induced by Metric",
"Metric Space is T2",
"Category:Normed Vector Spaces",
"Category:Hausdorff Spaces"
] |
proofwiki-19995 | Scalar Multiplication on Normed Vector Space is Continuous | Let $\struct { K, +, \circ_K }$ be a normed division ring with norm $\norm {\, \cdot \,}_K$.
Let $\struct {X, \norm {\, \cdot \,}_X }$ be a normed vector space over $K$.
Let $\struct {K \times X, \norm {\, \cdot \,}_P }$ be the direct product of $K$ and $X$ with the direct product norm $\norm {\, \cdot \,}_P$.
Let $\c... | Let $\tuple { \lambda_0 , x_0 } \in K \times X$.
Let $\epsilon' \in \R_{>0}$.
Set $\epsilon = \map \min { \epsilon', 1 }$
To show that $\circ$ is continuous, let $\tuple { \lambda , x } \in K \times X$ such that $\norm { \lambda_0 - \lambda }_K < \dfrac \epsilon { 1 + \norm { \lambda_0 }_K + \norm { x_0 }_X }$, and $\n... | Let $\struct { K, +, \circ_K }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Norm on Division Ring|norm]] $\norm {\, \cdot \,}_K$.
Let $\struct {X, \norm {\, \cdot \,}_X }$ be a [[Definition:Normed Vector Space|normed vector space]] over $K$.
Let $\struct {K \times X, \norm {\, \cdo... | Let $\tuple { \lambda_0 , x_0 } \in K \times X$.
Let $\epsilon' \in \R_{>0}$.
Set $\epsilon = \map \min { \epsilon', 1 }$
To show that $\circ$ is [[Definition:Continuous Mapping (Normed Vector Space)|continuous]], let $\tuple { \lambda , x } \in K \times X$ such that $\norm { \lambda_0 - \lambda }_K < \dfrac \epsilo... | Scalar Multiplication on Normed Vector Space is Continuous | https://proofwiki.org/wiki/Scalar_Multiplication_on_Normed_Vector_Space_is_Continuous | https://proofwiki.org/wiki/Scalar_Multiplication_on_Normed_Vector_Space_is_Continuous | [
"Continuous Mappings on Normed Vector Spaces",
"Scalar Multiplication"
] | [
"Definition:Normed Division Ring",
"Definition:Norm/Division Ring",
"Definition:Normed Vector Space",
"Definition:Direct Product of Vector Spaces/Finite Case",
"Definition:Direct Product Norm",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)"
] | [
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Direct Product Norm",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] |
proofwiki-19996 | Normed Vector Space is Hausdorff Topological Vector Space | Let $\struct { K, +_K, \circ_K }$ be a valued field with norm $\norm {\,\cdot\,}_K$.
Let $\struct {X, \norm {\, \cdot \,}_X }$ be a normed vector space over $K$.
Let $\tau$ be the topology on $X$ that consists of all open sets in $X$.
That is, $U \in \tau$ {{iff}}:
:$\forall x \in U: \exists \epsilon \in \R_{>0}: \map... | By its definition, a valued field is a normed division ring.
Let $\tau_K$ be the topology on $K$ induced by the metric induced by the norm $\norm {\,\cdot\,}_K$
From Normed Division Ring Operations are Continuous:Corollary, it follows that $\struct {K, \tau_K}$ is a topological field.
Let $d_X : X \times X \to \R_{\ge ... | Let $\struct { K, +_K, \circ_K }$ be a [[Definition:Valued Field|valued field]] with [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}_K$.
Let $\struct {X, \norm {\, \cdot \,}_X }$ be a [[Definition:Normed Vector Space|normed vector space]] over $K$.
Let $\tau$ be the [[Definition:Topology|topology]] on $... | By its definition, a [[Definition:Valued Field|valued field]] is a [[Definition:Normed Division Ring|normed division ring]].
Let $\tau_K$ be the [[Definition:Topology Induced by Metric|topology on $K$ induced]] by the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}_... | Normed Vector Space is Hausdorff Topological Vector Space | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff_Topological_Vector_Space | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff_Topological_Vector_Space | [
"Topological Vector Spaces",
"Hausdorff Topological Vector Spaces",
"Normed Vector Spaces",
"Hausdorff Topological Vector Spaces"
] | [
"Definition:Valued Field",
"Definition:Norm/Division Ring",
"Definition:Normed Vector Space",
"Definition:Topology",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Hausdorff Topological Vector Space"
] | [
"Definition:Valued Field",
"Definition:Normed Division Ring",
"Definition:Topology Induced by Metric",
"Definition:Metric Induced by Norm on Division Ring",
"Normed Division Ring Operations are Continuous/Corollary",
"Definition:Topological Field",
"Definition:Metric Induced by Norm",
"Definition:Inne... |
proofwiki-19997 | Hilbert Space is Hausdorff Topological Vector Space | Let $\struct {H, \innerprod \cdot \cdot}$ be a Hilbert space over $\mathbb F \in \set {\R, \C}$.
Let $d: H \times H \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}$ on $H$.
Let $\tau$ be the the topology on $H$ induced by the metric $d$.
Then $\struct {H, \tau}$ is a Hausdorff topolog... | If $\mathbb F = \R$, then Real Numbers form Field shows that $\mathbb F$ is a field.
From Euclidean Space is Normed Vector Space, it follows for $r \in \mathbb F$ that:
:$\size r = \sqrt {r^2}$
is a norm on $\mathbb F$.
If $\mathbb F = \C$, then Complex Numbers form Field shows that $\mathbb F$ is a field.
From Complex... | Let $\struct {H, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\mathbb F \in \set {\R, \C}$.
Let $d: H \times H \to \R_{\ge 0}$ be the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:Inner Product Norm|inner product norm]] $\norm {\,\cdot\,}$ on $H$.
Let $\tau... | If $\mathbb F = \R$, then [[Real Numbers form Field]] shows that $\mathbb F$ is a [[Definition:Field (Abstract Algebra)|field]].
From [[Euclidean Space is Normed Vector Space]], it follows for $r \in \mathbb F$ that:
:$\size r = \sqrt {r^2}$
is a [[Definition:Norm on Vector Space|norm]] on $\mathbb F$.
If $\mathbb ... | Hilbert Space is Hausdorff Topological Vector Space | https://proofwiki.org/wiki/Hilbert_Space_is_Hausdorff_Topological_Vector_Space | https://proofwiki.org/wiki/Hilbert_Space_is_Hausdorff_Topological_Vector_Space | [
"Hilbert Spaces",
"Hausdorff Topological Vector Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Metric Induced by Norm",
"Definition:Inner Product Norm",
"Definition:Topology Induced by Metric",
"Definition:Hausdorff Topological Vector Space"
] | [
"Real Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Euclidean Space is Normed Vector Space",
"Definition:Norm/Vector Space",
"Complex Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Complex Modulus is Norm",
"Definition:Norm/Vector Space",
"Definition:Valued Field",
"De... |
proofwiki-19998 | Furstenberg Topology is Topology | Let $\struct {\Z, \tau}$ be the topological space formed by the Furstenberg topology on the set of integers $\Z$.
Then $\tau$ is indeed a topology on $\Z$. | Recall the definition of the Furstenberg topology:
{{:Definition:Furstenberg Topology}}{{qed|lemma}}
In view of Union from Synthetic Basis is Topology it suffices to show that $\BB$ is a synthetic basis on $\Z$.
Recall the definition of synthetic basis:
{{Definition:Synthetic Basis/Definition 1}} | Let $\struct {\Z, \tau}$ be the [[Definition:Topological Space|topological space]] formed by the [[Definition:Furstenberg Topology|Furstenberg topology]] on the [[Definition:Integer|set of integers]] $\Z$.
Then $\tau$ is indeed a [[Definition:Topology|topology]] on $\Z$. | Recall the definition of the [[Definition:Furstenberg Topology|Furstenberg topology]]:
{{:Definition:Furstenberg Topology}}{{qed|lemma}}
In view of [[Union from Synthetic Basis is Topology]] it suffices to show that $\BB$ is a [[Definition:Synthetic Basis|synthetic basis]] on $\Z$.
Recall the definition of [[Definit... | Furstenberg Topology is Topology | https://proofwiki.org/wiki/Furstenberg_Topology_is_Topology | https://proofwiki.org/wiki/Furstenberg_Topology_is_Topology | [
"Furstenberg Topology"
] | [
"Definition:Topological Space",
"Definition:Furstenberg Topology",
"Definition:Integer",
"Definition:Topology"
] | [
"Definition:Furstenberg Topology",
"Union from Synthetic Basis is Topology",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Synthetic Basis/Definition 1"
] |
proofwiki-19999 | Equivalence of Definitions of Hilbert Space | Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.
{{TFAE|def = Hilbert Space|view = Hilbert space}}
=== Definition 1===
{{:Definition:Hilbert Space/Definition 1}}
=== Definition 2===
{{:Definition:Hilbert Space/Definition 2}} | === Definition 1 implies Definition 2 ===
By definition of complete metric space, every Cauchy sequence in $H$ is convergent.
By definition of Banach space, it follows that $\struct { H, \norm {\,\cdot\,}_H }$ is a Banach space.
{{qed|lemma}} | Let $H$ be a [[Definition:Vector Space|vector space]] over $\mathbb F \in \set {\R, \C}$.
{{TFAE|def = Hilbert Space|view = Hilbert space}}
=== [[Definition:Hilbert Space/Definition 1|Definition 1]]===
{{:Definition:Hilbert Space/Definition 1}}
=== [[Definition:Hilbert Space/Definition 2|Definition 2]]===
{{:Definitio... | === [[Definition:Hilbert Space/Definition 1|Definition 1]] implies [[Definition:Hilbert Space/Definition 2|Definition 2]] ===
By definition of [[Definition:Complete Metric Space|complete metric space]], every [[Definition:Cauchy Sequence|Cauchy sequence]] in $H$ is [[Definition:Convergent Sequence (Metric Space)|conve... | Equivalence of Definitions of Hilbert Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Hilbert_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Hilbert_Space | [
"Hilbert Spaces"
] | [
"Definition:Vector Space",
"Definition:Hilbert Space/Definition 1",
"Definition:Hilbert Space/Definition 2"
] | [
"Definition:Hilbert Space/Definition 1",
"Definition:Hilbert Space/Definition 2",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Banach Space",
"Definition:Banach Space",
"Definition:Hilbert Space/Definition 2",
"Definitio... |
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