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-6500 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i z}
| r = \cos z + i \sin z
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i z}
| r = \cos z - i \sin z
| c = {{Corollary|Euler's Formula}}
}}
{{eqn | ll= \leadsto
| l = e^{i z} - e^{-i z}
| r = \paren {\cos z + i \sin z} - \paren {... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i z}
| r = \cos z + i \sin z
| c = [[Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i z}
| r = \cos z - i \sin z
| c = {{Corollary|Euler's Formula}}
}}
{{eqn | ll= \leadsto
| l = e^{i z} - e^{-i z}
| r = \paren {\cos z + i \sin z} - \par... | Euler's Sine Identity/Proof 3 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Proof_3 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Euler's Formula"
] |
proofwiki-6501 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall the definition of the sine function:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots + \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + ... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall the definition of the [[Definition:Real Sine Function|sine function]]:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots + \paren {-1}^n \... | Euler's Sine Identity/Real Domain/Proof 1 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_1 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Definition:Sine/Real Function",
"Definition:Exponential Function/Real/Power Series Expansion",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-6502 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall Euler's Formula:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} - e^{-i x} }{2 i}
| r = \frac {\paren {\cos x + i \sin x} - \paren {\map \cos {-x} + i \map \sin {-x} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos x + i \sin x - \cos x - i \map \sin {-... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall [[Euler's Formula/Real Domain|Euler's Formula]]:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} - e^{-i x} }{2 i}
| r = \frac {\paren {\cos x + i \sin x} - \paren {\map \cos {-x} + i \map \sin {-x} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos x ... | Euler's Sine Identity/Real Domain/Proof 2 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_2 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-6503 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = Euler's Formula: Corollary
}}
{{eqn | ll= \leadsto
| l = e^{i x} - e^{-i x}
| r = \paren {\cos x + i \sin x} - \paren {\co... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = [[Euler's Formula/Real Domain|Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = [[Euler's Formula/Real Domain/Corollary|Euler's Formula: Corollary]]
}}
{{eqn | ll= \leadsto
| l... | Euler's Sine Identity/Real Domain/Proof 3 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_3 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Euler's Formula/Real Domain/Corollary"
] |
proofwiki-6504 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b} + i \, \map \sin {a + b}
| r = e^{i \paren {a + b} }
| c = Euler's Formula
}}
{{eqn | r = e^{i a} e^{i b}
| c = Exponential of Sum
}}
{{eqn | r = \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}
| c = Euler's Formula
}}
{{eqn | r = \paren {\cos a ... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b} + i \, \map \sin {a + b}
| r = e^{i \paren {a + b} }
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{i a} e^{i b}
| c = [[Exponential of Sum]]
}}
{{eqn | r = \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}
| c = [[Euler's Formula]]
}}
{{eqn | r = \pa... | Sine of Sum/Proof 1 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_1 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Euler's Formula",
"Exponential of Sum",
"Euler's Formula",
"Complex Numbers form Field",
"Definition:Complex Number/Imaginary Part"
] |
proofwiki-6505 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | Recall the analytic definitions of sine and cosine:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Let:
{{begin-eqn}}
{{eqn | l = \map g a
| r = \map \sin {a + b} - \sin a... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | Recall the analytic definitions of [[Definition:Complex Sine Function|sine]] and [[Definition:Complex Cosine Function|cosine]]:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
... | Sine of Sum/Proof 2 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_2 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Definition:Sine/Complex Function",
"Definition:Cosine/Complex Function",
"Definition:Derivative/Real Function/With Respect To",
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Derivative of Constant",
"Square of Real Number is Non-Negative"
] |
proofwiki-6506 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \sin a \cos b + \cos a \sin b
| r = \paren {\frac {e^{i a} - e^{-i a} }{2 i} } \cos b + \cos a \paren {\frac {e^{i b} - e^{-i b} }{2 i} }
| c = Euler's Sine Identity
}}
{{eqn | r = \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} + e^{-i b} } 2} + \paren {\frac {... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \sin a \cos b + \cos a \sin b
| r = \paren {\frac {e^{i a} - e^{-i a} }{2 i} } \cos b + \cos a \paren {\frac {e^{i b} - e^{-i b} }{2 i} }
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} + e^{-i b} } 2} + \paren {\fr... | Sine of Sum/Proof 3 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_3 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Exponential of Sum",
"Euler's Sine Identity"
] |
proofwiki-6507 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \sin \left({a + b}\right)
| r = \cos \left({\frac \pi 2 - \left({a + b}\right)}\right)
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \cos \left({\left({\frac \pi 2 - a}\right) - b}\right)
}}
{{eqn | r = \cos \left({\frac \pi 2 - a}\right) \cos b + \sin \left({\frac \pi 2 - ... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | l = \sin \left({a + b}\right)
| r = \cos \left({\frac \pi 2 - \left({a + b}\right)}\right)
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = \cos \left({\left({\frac \pi 2 - a}\right) - b}\right)
}}
{{eqn | r = \cos \left({\frac \pi 2 - a}\right) \cos b + \sin \left({\frac \pi ... | Sine of Sum/Proof 4 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_4 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Cosine of Complement equals Sine",
"Cosine of Difference",
"Cosine of Complement equals Sine",
"Sine of Complement equals Cosine"
] |
proofwiki-6508 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | n = 1
| l = 2 \sin a \cos b
| r = \sin \paren {a + b} + \sin \paren {a - b}
| c = Werner Formula for Sine by Cosine: Proof 2
}}
{{eqn | n = 2
| l = 2 \cos a \sin b
| r = \sin \paren {a + b} - \sin \paren {a - b}
| c = Werner Formula for Cosine by Sine: Proof 2
}... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | {{begin-eqn}}
{{eqn | n = 1
| l = 2 \sin a \cos b
| r = \sin \paren {a + b} + \sin \paren {a - b}
| c = [[Werner Formulas/Sine by Cosine/Proof 2|Werner Formula for Sine by Cosine: Proof 2]]
}}
{{eqn | n = 2
| l = 2 \cos a \sin b
| r = \sin \paren {a + b} - \sin \paren {a - b}
| c = [... | Sine of Sum/Proof 5 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_5 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Werner Formulas/Sine by Cosine/Proof 2",
"Werner Formulas/Cosine by Sine/Proof 2"
] |
proofwiki-6509 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | :900px
We begin by enclosing a right-angled triangle $BEF$ with hypotenuse $BF$ of length $1$, inside rectangle $ABCD$.
Let $\angle EBF = a$ and $\angle ABE = b$.
Therefore:
{{begin-eqn}}
{{eqn | l = BF
| r = 1
| c = Given
}}
{{eqn | l = BE
| r = \cos a
| c = {{Defof|Cosine of Angle}}
}}
{{eqn |... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | :[[File:Angle-sum.png|900px]]
We begin by enclosing a [[Definition:Right-Angled Triangle|right-angled triangle]] $BEF$ with [[Definition:Hypotenuse|hypotenuse]] $BF$ of length $1$, inside [[Definition:Rectangle|rectangle]] $ABCD$.
Let $\angle EBF = a$ and $\angle ABE = b$.
Therefore:
{{begin-eqn}}
{{eqn | l = BF
... | Sine of Sum/Proof 6 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_6 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"File:Angle-sum.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Quadrilateral/Rectangle"
] |
proofwiki-6510 | Sine of Sum | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | 350px
Let two triangles $\triangle ABC$ and $\triangle ADC$ be inscribed in a circle on opposite
sides of diameter $AC$.
By Thales' Theorem, they are both right triangles and $\angle ADC$ and $\angle ABC$ are right angles.
Let the diameter $AC = 1$.
Let $\angle DAC = \alpha$ and $\angle CAB = \beta$.
From the construct... | :$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$ | [[File:Sum of Sines using Ptolemy.png|350px]]
Let two [[Definition:Triangle (Geometry)|triangles]] $\triangle ABC$ and $\triangle ADC$ be [[Definition:Angle Inscribed in Circle|inscribed]] in a [[Definition:Circle|circle]] on [[Definition:Opposite Sides|opposite]]
[[Definition:Side of Polygon|sides]] of [[Definition:D... | Sine of Sum/Proof 7 | https://proofwiki.org/wiki/Sine_of_Sum | https://proofwiki.org/wiki/Sine_of_Sum/Proof_7 | [
"Sine of Sum",
"Sine Function",
"Trigonometric Addition Formulas"
] | [] | [
"File:Sum of Sines using Ptolemy.png",
"Definition:Triangle (Geometry)",
"Definition:Angle Inscribed in Circle",
"Definition:Circle",
"Definition:Polygon/Opposite",
"Definition:Polygon/Side",
"Definition:Circle/Diameter",
"Thales' Theorem",
"Definition:Triangle (Geometry)/Right-Angled",
"Definitio... |
proofwiki-6511 | Hyperbolic Sine Function is Odd | :$\map \sinh {-x} = -\sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = \frac {e^{-x} - e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{-x} - e^x} 2
}}
{{eqn | r = -\frac {e^x - e^{-x} } 2
}}
{{eqn | r = -\sinh x
}}
{{end-eqn}}
{{qed}} | :$\map \sinh {-x} = -\sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = \frac {e^{-x} - e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{-x} - e^x} 2
}}
{{eqn | r = -\frac {e^x - e^{-x} } 2
}}
{{eqn | r = -\sinh x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Sine Function is Odd/Proof 1 | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd/Proof_1 | [
"Hyperbolic Sine Function",
"Hyperbolic Sine Function is Odd",
"Examples of Odd Functions"
] | [] | [] |
proofwiki-6512 | Hyperbolic Sine Function is Odd | :$\map \sinh {-x} = -\sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = -i \, \map \sin {-i x}
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = i \, \map \sin {i x}
| c = Sine Function is Odd
}}
{{eqn | r = -\sinh x
| c = Hyperbolic Sine in terms of Sine
}}
{{end-eqn}}
{{qed}} | :$\map \sinh {-x} = -\sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = -i \, \map \sin {-i x}
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = i \, \map \sin {i x}
| c = [[Sine Function is Odd]]
}}
{{eqn | r = -\sinh x
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{end-eqn}}
{{qed}} | Hyperbolic Sine Function is Odd/Proof 2 | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd/Proof_2 | [
"Hyperbolic Sine Function",
"Hyperbolic Sine Function is Odd",
"Examples of Odd Functions"
] | [] | [
"Hyperbolic Sine in terms of Sine",
"Sine Function is Odd",
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-6513 | Exponential of Sum/Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\exp z$ be the exponential of $z$.
Then:
:$\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$ | This proof is based on the definition of the complex exponential as the unique solution of the differential equation:
:$\dfrac \d {\d z} \exp = \exp$
which satisfies the initial condition $\map \exp 0 = 1$.
Define the complex function $f: \C \to \C$ by:
:$\map f z = \map \exp z \, \map \exp {z_1 + z_2 - z}$
Then find ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\exp z$ be the [[Definition:Complex Exponential Function|exponential of $z$]].
Then:
:$\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$ | This proof is based on the [[Definition:Exponential Function/Complex/Differential Equation|definition of the complex exponential]] as the [[Definition:Unique|unique]] [[Definition:Solution of Differential Equation|solution]] of the [[Definition:Differential Equation|differential equation]]:
:$\dfrac \d {\d z} \exp = \... | Exponential of Sum/Complex Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers | [
"Exponential of Sum"
] | [
"Definition:Complex Number",
"Definition:Exponential Function/Complex"
] | [
"Definition:Exponential Function/Complex/Differential Equation",
"Definition:Unique",
"Definition:Differential Equation/Solution",
"Definition:Differential Equation",
"Definition:Initial Condition",
"Definition:Complex Function",
"Definition:Derivative/Complex Function",
"Derivative of Complex Composi... |
proofwiki-6514 | Hyperbolic Cosine Function is Even | :$\map \cosh {-x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \frac {e^{-x} + e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{-x} + e^x} 2
}}
{{eqn | r = \frac {e^x + e^{-x} } 2
}}
{{eqn | r = \cosh x
}}
{{end-eqn}}
{{qed}} | :$\map \cosh {-x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \frac {e^{-x} + e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{-x} + e^x} 2
}}
{{eqn | r = \frac {e^x + e^{-x} } 2
}}
{{eqn | r = \cosh x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine Function is Even/Proof 1 | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even/Proof_1 | [
"Hyperbolic Cosine Function",
"Hyperbolic Cosine Function is Even",
"Examples of Even Functions"
] | [] | [] |
proofwiki-6515 | Hyperbolic Cosine Function is Even | :$\map \cosh {-x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \map \cos {-i x}
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \map \cos {i x}
| c = Cosine Function is Even
}}
{{eqn | r = \cosh x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \cosh {-x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \map \cos {-i x}
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \map \cos {i x}
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \cosh x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine Function is Even/Proof 2 | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even/Proof_2 | [
"Hyperbolic Cosine Function",
"Hyperbolic Cosine Function is Even",
"Examples of Even Functions"
] | [] | [
"Hyperbolic Cosine in terms of Cosine",
"Cosine Function is Even",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-6516 | Hyperbolic Tangent Function is Odd | Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers.
Then $\tanh$ is odd:
:$\map \tanh {-x} = -\tanh x$ | {{begin-eqn}}
{{eqn | l = \map \tanh {-x}
| r = \frac {\map \sinh {-x} } {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {-\sinh x} {\map \cosh {-x} }
| c = Hyperbolic Sine Function is Odd
}}
{{eqn | r = \frac {-\sinh x} {\cosh x}
| c = Hyperbolic Cosine F... | Let $\tanh: \C \to \C$ be the [[Definition:Hyperbolic Tangent|hyperbolic tangent function]] on the [[Definition:Complex Number|set of complex numbers]].
Then $\tanh$ is [[Definition:Odd Function|odd]]:
:$\map \tanh {-x} = -\tanh x$ | {{begin-eqn}}
{{eqn | l = \map \tanh {-x}
| r = \frac {\map \sinh {-x} } {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {-\sinh x} {\map \cosh {-x} }
| c = [[Hyperbolic Sine Function is Odd]]
}}
{{eqn | r = \frac {-\sinh x} {\cosh x}
| c = [[Hyperbolic Co... | Hyperbolic Tangent Function is Odd | https://proofwiki.org/wiki/Hyperbolic_Tangent_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Tangent_Function_is_Odd | [
"Hyperbolic Tangent Function",
"Examples of Odd Functions"
] | [
"Definition:Hyperbolic Tangent",
"Definition:Complex Number",
"Definition:Odd Function"
] | [
"Hyperbolic Sine Function is Odd",
"Hyperbolic Cosine Function is Even"
] |
proofwiki-6517 | Hyperbolic Sine in terms of Sine | Let $z \in \C$ be a complex number.
Then:
:$i \sinh z = \map \sin {i z}$
where:
:$\sin$ denotes the complex sine
:$\sinh$ denotes the hyperbolic sine
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \sin {i z}
| r = \frac {e^{i \paren {i z} } - e^{i \paren {-i z} } } {2 i}
| c = Euler's Sine Identity
}}
{{eqn | r = \paren {-i} \frac {e^{-z} - e^z} 2
| c = $i^2 = -1$
}}
{{eqn | r = i \frac {e^z - e^{-z} } 2
| c = $i^2 = -1$
}}
{{eqn | r = i \sinh z
| c = ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \sinh z = \map \sin {i z}$
where:
:$\sin$ denotes the [[Definition:Complex Sine Function|complex sine]]
:$\sinh$ denotes the [[Definition:Hyperbolic Sine|hyperbolic sine]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \sin {i z}
| r = \frac {e^{i \paren {i z} } - e^{i \paren {-i z} } } {2 i}
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \paren {-i} \frac {e^{-z} - e^z} 2
| c = $i^2 = -1$
}}
{{eqn | r = i \frac {e^z - e^{-z} } 2
| c = $i^2 = -1$
}}
{{eqn | r = i \sinh z
| ... | Hyperbolic Sine in terms of Sine | https://proofwiki.org/wiki/Hyperbolic_Sine_in_terms_of_Sine | https://proofwiki.org/wiki/Hyperbolic_Sine_in_terms_of_Sine | [
"Sine Function",
"Hyperbolic Sine Function"
] | [
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Hyperbolic Sine",
"Definition:Complex Number/Imaginary Unit"
] | [
"Euler's Sine Identity"
] |
proofwiki-6518 | Hyperbolic Cosine in terms of Cosine | Let $z \in \C$ be a complex number.
Then:
:$\cosh z = \map \cos {i z}$
where:
:$\cos$ denotes the complex cosine
:$\cosh$ denotes the hyperbolic cosine
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \cos {i z}
| r = \frac {e^{i \paren {i z} } + e^{-i \paren {i z} } } 2
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac {e^{-z} + e^z} 2
| c = $i^2 = -1$
}}
{{eqn | r = \cosh z
| c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\cosh z = \map \cos {i z}$
where:
:$\cos$ denotes the [[Definition:Complex Cosine Function|complex cosine]]
:$\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i^2 ... | {{begin-eqn}}
{{eqn | l = \map \cos {i z}
| r = \frac {e^{i \paren {i z} } + e^{-i \paren {i z} } } 2
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac {e^{-z} + e^z} 2
| c = $i^2 = -1$
}}
{{eqn | r = \cosh z
| c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine in terms of Cosine | https://proofwiki.org/wiki/Hyperbolic_Cosine_in_terms_of_Cosine | https://proofwiki.org/wiki/Hyperbolic_Cosine_in_terms_of_Cosine | [
"Cosine Function",
"Hyperbolic Cosine Function"
] | [
"Definition:Complex Number",
"Definition:Cosine/Complex Function",
"Definition:Hyperbolic Cosine",
"Definition:Complex Number/Imaginary Unit"
] | [
"Euler's Cosine Identity"
] |
proofwiki-6519 | Hyperbolic Tangent in terms of Tangent | Let $z \in \C$ be a complex number.
Then:
:$i \tanh z = \map \tan {i z}$
where:
:$\tan$ denotes the tangent function
:$\tanh$ denotes the hyperbolic tangent
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \tan {i z}
| r = \frac {\map \sin {i z} } {\map \cos {i z} }
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {i \sinh z} {\map \cos {i z} }
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = \frac {i \sinh z} {\cosh z}
| c = Hyperbolic Cosine in t... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \tanh z = \map \tan {i z}$
where:
:$\tan$ denotes the [[Definition:Complex Tangent Function|tangent function]]
:$\tanh$ denotes the [[Definition:Hyperbolic Tangent|hyperbolic tangent]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]... | {{begin-eqn}}
{{eqn | l = \map \tan {i z}
| r = \frac {\map \sin {i z} } {\map \cos {i z} }
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {i \sinh z} {\map \cos {i z} }
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = \frac {i \sinh z} {\cosh z}
| c = [[Hyperbolic Cosin... | Hyperbolic Tangent in terms of Tangent | https://proofwiki.org/wiki/Hyperbolic_Tangent_in_terms_of_Tangent | https://proofwiki.org/wiki/Hyperbolic_Tangent_in_terms_of_Tangent | [
"Tangent Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Complex Number",
"Definition:Tangent Function/Complex",
"Definition:Hyperbolic Tangent",
"Definition:Complex Number/Imaginary Unit"
] | [
"Hyperbolic Sine in terms of Sine",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-6520 | Order-Extension Principle | Let $S$ be a set.
Let $\preceq$ be an ordering on $S$.
Then there exists a total ordering $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\preceq$ be an ordering on the set $S$.
If $\preceq$ is a total ordering, the result is complete.
Suppose, then, that $\preceq$ is not a total ordering.
Let $\TT$ be the set of orderings on $S$ that extend $\preceq$, ordered by inclusion.
Let $C$ be a chain in $T$.
By Union of Chain of Orderings is Ordering, $\big... | Let $S$ be a [[Definition:Set|set]].
Let $\preceq$ be an [[Definition:Ordering|ordering]] on $S$.
Then there exists a [[Definition:Total Ordering|total ordering]] $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\preceq$ be an [[Definition:Ordering|ordering]] on the [[Definition:Set|set]] $S$.
If $\preceq$ is a [[Definition:Total Ordering|total ordering]], the result is complete.
Suppose, then, that $\preceq$ is not a [[Definition:Total Ordering|total ordering]].
Let $\TT$ be the [[Definition:Set|set]] of [[Definition... | Order-Extension Principle/Proof 1 | https://proofwiki.org/wiki/Order-Extension_Principle | https://proofwiki.org/wiki/Order-Extension_Principle/Proof_1 | [
"Set Theory",
"Order Theory",
"Named Theorems",
"Order-Extension Principle"
] | [
"Definition:Set",
"Definition:Ordering",
"Definition:Total Ordering"
] | [
"Definition:Ordering",
"Definition:Set",
"Definition:Total Ordering",
"Definition:Total Ordering",
"Definition:Set",
"Definition:Ordering",
"Definition:Extension of Relation",
"Subset Relation is Ordering",
"Definition:Chain (Order Theory)/Subset Relation",
"Union of Chain of Orderings is Ordering... |
proofwiki-6521 | Order-Extension Principle | Let $S$ be a set.
Let $\preceq$ be an ordering on $S$.
Then there exists a total ordering $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\prec$ be the reflexive reduction of $\preceq$.
By Reflexive Reduction of Ordering is Strict Ordering, $\prec$ is a strict ordering.
By the strict form of the Order-Extension Principle, there exists a strict total ordering $<$ on $S$ such that:
:$\forall a, b \in S: \paren {a \prec b \implies a < b}$
Let $\le$ be ... | Let $S$ be a [[Definition:Set|set]].
Let $\preceq$ be an [[Definition:Ordering|ordering]] on $S$.
Then there exists a [[Definition:Total Ordering|total ordering]] $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\prec$ be the [[Definition:Reflexive Reduction|reflexive reduction]] of $\preceq$.
By [[Reflexive Reduction of Ordering is Strict Ordering]], $\prec$ is a [[Definition:Strict Ordering|strict ordering]].
By the [[Order-Extension Principle/Strict|strict form of the Order-Extension Principle]], there exists a [[Def... | Order-Extension Principle/Proof 2 | https://proofwiki.org/wiki/Order-Extension_Principle | https://proofwiki.org/wiki/Order-Extension_Principle/Proof_2 | [
"Set Theory",
"Order Theory",
"Named Theorems",
"Order-Extension Principle"
] | [
"Definition:Set",
"Definition:Ordering",
"Definition:Total Ordering"
] | [
"Definition:Reflexive Reduction",
"Reflexive Reduction of Ordering is Strict Ordering",
"Definition:Strict Ordering",
"Order-Extension Principle/Strict",
"Definition:Strict Total Ordering",
"Definition:Reflexive Closure",
"Law of Excluded Middle",
"Definition:Reflexive Closure",
"Reflexive Closure o... |
proofwiki-6522 | Order-Extension Principle | Let $S$ be a set.
Let $\preceq$ be an ordering on $S$.
Then there exists a total ordering $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\AA$ be the set of relations $A$ on $S$ with the property that the transitive closure $A^+$ of $A$ is a strict ordering of $S$.
For each $\tuple {x, y} \in S \times S$, let $\tuple {x, y}' = \tuple {y, x}$.
Let $A \in \AA$.
Let $\tuple {x, y} \in S \times S$.
Let $\tuple {x, y} \in A^+$.
Then:
:$\paren {A \cup \se... | Let $S$ be a [[Definition:Set|set]].
Let $\preceq$ be an [[Definition:Ordering|ordering]] on $S$.
Then there exists a [[Definition:Total Ordering|total ordering]] $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | Let $\AA$ be the [[Definition:Set|set]] of [[Definition:Endorelation|relations]] $A$ on $S$ with the property that the [[Definition:Transitive Closure of Relation|transitive closure]] $A^+$ of $A$ is a [[Definition:Strict Ordering|strict ordering]] of $S$.
For each $\tuple {x, y} \in S \times S$, let $\tuple {x, y}' =... | Order-Extension Principle/Strict/Proof 1 | https://proofwiki.org/wiki/Order-Extension_Principle | https://proofwiki.org/wiki/Order-Extension_Principle/Strict/Proof_1 | [
"Set Theory",
"Order Theory",
"Named Theorems",
"Order-Extension Principle"
] | [
"Definition:Set",
"Definition:Ordering",
"Definition:Total Ordering"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Transitive Closure of Relation",
"Definition:Strict Ordering",
"Definition:Non-Comparable Elements",
"Strict Ordering can be Expanded to Compare Additional Pair",
"Definition:Finite Subset",
"Definition:Strict Ordering",
"Definition:Asymmetric... |
proofwiki-6523 | Order-Extension Principle | Let $S$ be a set.
Let $\preceq$ be an ordering on $S$.
Then there exists a total ordering $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | For the purposes of this proof, a relation $<_U$ on a subset $U$ of $S$ will be considered '''compatible''' with $\prec$ {{iff}}:
:$\forall a, b \in U: a \prec b \implies a < b$
Let $M$ be the set of partial mappings $f$ from $S \times S$ to $\left\{ {0, 1}\right\}$ such that for all $x, y, z \in S$:
:$(a): \quad \left... | Let $S$ be a [[Definition:Set|set]].
Let $\preceq$ be an [[Definition:Ordering|ordering]] on $S$.
Then there exists a [[Definition:Total Ordering|total ordering]] $\le$ on $S$ such that:
:$\forall a, b \in S: \paren {a \preceq b \implies a \le b}$ | For the purposes of this proof, a [[Definition:Endorelation|relation]] $<_U$ on a [[Definition:Subset|subset]] $U$ of $S$ will be considered '''compatible''' with $\prec$ {{iff}}:
:$\forall a, b \in U: a \prec b \implies a < b$
Let $M$ be the [[Definition:Set|set]] of [[Definition:Partial Mapping|partial mappings]] $... | Order-Extension Principle/Strict/Proof 2 | https://proofwiki.org/wiki/Order-Extension_Principle | https://proofwiki.org/wiki/Order-Extension_Principle/Strict/Proof_2 | [
"Set Theory",
"Order Theory",
"Named Theorems",
"Order-Extension Principle"
] | [
"Definition:Set",
"Definition:Ordering",
"Definition:Total Ordering"
] | [
"Definition:Endorelation",
"Definition:Subset",
"Definition:Set",
"Definition:Many-to-One Relation",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Strict Total Ordering",
"Cowen-Engeler Lemma",
"Definition:Finite Set",
"Definition... |
proofwiki-6524 | Union of Chain of Orderings is Ordering | Let $S$ be a set.
Let $C$ be a non-empty chain of orderings on $S$.
Then $\bigcup C$ is an ordering on $S$. | {{improve|Use letters (mathcaled if you like) to denote an arbitrary ordering rather than actual arbitrary ordering symbols which make it confusing and difficult to read}}
Let $\preceq$ be an arbitrary element of $C$.
Let ${\sim} = \bigcup C$.
Checking in turn each of the criteria for an ordering:
Let $a, b \in S$.
Let... | Let $S$ be a [[Definition:Set|set]].
Let $C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain of Sets|chain]] of [[Definition:Ordering|orderings]] on $S$.
Then $\bigcup C$ is an [[Definition:Ordering|ordering]] on $S$. | {{improve|Use letters (mathcaled if you like) to denote an arbitrary ordering rather than actual arbitrary ordering symbols which make it confusing and difficult to read}}
Let $\preceq$ be an arbitrary [[Definition:Element|element]] of $C$.
Let ${\sim} = \bigcup C$.
Checking in turn each of the criteria for an [[De... | Union of Chain of Orderings is Ordering | https://proofwiki.org/wiki/Union_of_Chain_of_Orderings_is_Ordering | https://proofwiki.org/wiki/Union_of_Chain_of_Orderings_is_Ordering | [
"Set Theory",
"Order Theory"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Ordering",
"Definition:Ordering"
] | [
"Definition:Element",
"Definition:Ordering",
"Definition:Chain (Order Theory)",
"Definition:Ordering",
"Definition:Antisymmetric Relation",
"Definition:Ordering",
"Definition:Element",
"Definition:Chain (Order Theory)",
"Definition:Ordering",
"Definition:Antisymmetric Relation",
"Definition:Orde... |
proofwiki-6525 | Partial Ordering can be Expanded to compare Additional Pair | Let $\struct {S, \preceq}$ be an ordered set.
Let $a$ and $b$ be non-comparable elements of $S$.
That is, let:
:$a \not\preceq b$
and:
:$b \not\preceq a$
Let ${\preceq'} = {\preceq} \cup \set {\tuple {a, b} }$.
Let $\preceq'^+$ be the transitive closure of $\preceq'$.
Then:
:$\preceq'^+$ is an ordering.
$\preceq'^+$ ca... | $\preceq'^+$ is a superset of $\preceq$.
By Relation Contains Diagonal Relation iff Reflexive:
:$\Delta_S \subseteq \preceq$
where $\Delta_S$ denotes the diagonal relation.
By Subset Relation is Transitive it follows that:
:$\Delta_S \subseteq \preceq'^+$
By Relation Contains Diagonal Relation iff Reflexive, it follows... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $a$ and $b$ be [[Definition:Non-Comparable Elements|non-comparable elements]] of $S$.
That is, let:
:$a \not\preceq b$
and:
:$b \not\preceq a$
Let ${\preceq'} = {\preceq} \cup \set {\tuple {a, b} }$.
Let $\preceq'^+$ be the [[Definition:T... | $\preceq'^+$ is a [[Definition:Superset|superset]] of $\preceq$.
By [[Relation Contains Diagonal Relation iff Reflexive]]:
:$\Delta_S \subseteq \preceq$
where $\Delta_S$ denotes the [[Definition:Diagonal Relation|diagonal relation]].
By [[Subset Relation is Transitive]] it follows that:
:$\Delta_S \subseteq \preceq'^... | Partial Ordering can be Expanded to compare Additional Pair/Proof 1 | https://proofwiki.org/wiki/Partial_Ordering_can_be_Expanded_to_compare_Additional_Pair | https://proofwiki.org/wiki/Partial_Ordering_can_be_Expanded_to_compare_Additional_Pair/Proof_1 | [
"Order Theory",
"Partial Ordering can be Expanded to compare Additional Pair"
] | [
"Definition:Ordered Set",
"Definition:Non-Comparable Elements",
"Definition:Transitive Closure of Relation",
"Definition:Ordering"
] | [
"Definition:Subset/Superset",
"Equivalence of Definitions of Reflexive Relation",
"Definition:Diagonal Relation",
"Subset Relation is Transitive",
"Equivalence of Definitions of Reflexive Relation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Transitive Closure of Re... |
proofwiki-6526 | Partial Ordering can be Expanded to compare Additional Pair | Let $\struct {S, \preceq}$ be an ordered set.
Let $a$ and $b$ be non-comparable elements of $S$.
That is, let:
:$a \not\preceq b$
and:
:$b \not\preceq a$
Let ${\preceq'} = {\preceq} \cup \set {\tuple {a, b} }$.
Let $\preceq'^+$ be the transitive closure of $\preceq'$.
Then:
:$\preceq'^+$ is an ordering.
$\preceq'^+$ ca... | Let $\prec$ be the reflexive reduction of $\preceq$.
Let $\prec' = {\prec} \cup \set {\tuple {a, b} }$.
By Reflexive Reduction of Ordering is Strict Ordering, $\prec$ is a strict ordering.
Define a relation $\prec'_2$ by letting $p \prec'_2 q$ {{iff}}:
:$p \prec q$ or
:$p \preceq a$ and $b \preceq q$
By Strict Ordering... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $a$ and $b$ be [[Definition:Non-Comparable Elements|non-comparable elements]] of $S$.
That is, let:
:$a \not\preceq b$
and:
:$b \not\preceq a$
Let ${\preceq'} = {\preceq} \cup \set {\tuple {a, b} }$.
Let $\preceq'^+$ be the [[Definition:T... | Let $\prec$ be the [[Definition:Reflexive Reduction|reflexive reduction]] of $\preceq$.
Let $\prec' = {\prec} \cup \set {\tuple {a, b} }$.
By [[Reflexive Reduction of Ordering is Strict Ordering]], $\prec$ is a [[Definition:Strict Ordering|strict ordering]].
Define a relation $\prec'_2$ by letting $p \prec'_2 q$ {{i... | Partial Ordering can be Expanded to compare Additional Pair/Proof 2 | https://proofwiki.org/wiki/Partial_Ordering_can_be_Expanded_to_compare_Additional_Pair | https://proofwiki.org/wiki/Partial_Ordering_can_be_Expanded_to_compare_Additional_Pair/Proof_2 | [
"Order Theory",
"Partial Ordering can be Expanded to compare Additional Pair"
] | [
"Definition:Ordered Set",
"Definition:Non-Comparable Elements",
"Definition:Transitive Closure of Relation",
"Definition:Ordering"
] | [
"Definition:Reflexive Reduction",
"Reflexive Reduction of Ordering is Strict Ordering",
"Definition:Strict Ordering",
"Strict Ordering can be Expanded to Compare Additional Pair",
"Definition:Strict Ordering",
"Definition:Transitive Closure of Relation",
"Definition:Reflexive Closure",
"Definition:Ref... |
proofwiki-6527 | Recursive Construction of Transitive Closure | Let $\RR$ be a relation.
The '''transitive closure''' $\RR^+$ of $\RR$ can be constructed as follows:
Let:
:$\RR_n := \begin {cases} \RR & : n = 0 \\ \RR_{n - 1} \cup \set {\tuple {x_1, x_3}: \exists x_2: \tuple {x_1, x_2} \in \RR_{n-1} \land \tuple {x_2, x_3} \in \RR_{n - 1} } & : n > 0 \end{cases}$
Finally, let:
:$\d... | We must show that:
:$(1): \quad \RR \subseteq \RR^+$
:$(2): \quad \RR^+$ is transitive
:$(3): \quad \RR^+$ is the smallest relation with both of those characteristics. | Let $\RR$ be a [[Definition:Relation|relation]].
The '''[[Definition:Transitive Closure of Relation|transitive closure]]''' $\RR^+$ of $\RR$ can be constructed as follows:
Let:
:$\RR_n := \begin {cases} \RR & : n = 0 \\ \RR_{n - 1} \cup \set {\tuple {x_1, x_3}: \exists x_2: \tuple {x_1, x_2} \in \RR_{n-1} \land \tupl... | We must show that:
:$(1): \quad \RR \subseteq \RR^+$
:$(2): \quad \RR^+$ is [[Definition:Transitive Relation|transitive]]
:$(3): \quad \RR^+$ is the smallest [[Definition:Relation|relation]] with both of those characteristics. | Recursive Construction of Transitive Closure | https://proofwiki.org/wiki/Recursive_Construction_of_Transitive_Closure | https://proofwiki.org/wiki/Recursive_Construction_of_Transitive_Closure | [
"Transitive Closures"
] | [
"Definition:Relation",
"Definition:Transitive Closure of Relation",
"Definition:Transitive Closure of Relation"
] | [
"Definition:Transitive Relation",
"Definition:Relation",
"Definition:Transitive Relation",
"Definition:Transitive Relation"
] |
proofwiki-6528 | Contour Integral is Well-Defined | Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane $\C$.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ d... | Define $g_k: \closedint {a_k} {b_k} \to \C$ by $\map {g_k} t = \map f {\map {\gamma_k} t} \map {\gamma_k'} t$ for all $k \in \set {1, \ldots, n}$.
By definition of smooth path, it follows that $\gamma_k$ and $\gamma_k'$ are continuous for all $k \in \set {1, \ldots, n}$.
From Continuity of Composite Mapping/Corollary a... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour]] defined by a [[Definition:Finite Sequence|finite sequence]] $C_1, \ldots, C_n$ of [[Definition:Directed Smooth Curve|directed smooth curves]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Let $C_k$ be [[Definition:Parameterization of Directed Smooth... | Define $g_k: \closedint {a_k} {b_k} \to \C$ by $\map {g_k} t = \map f {\map {\gamma_k} t} \map {\gamma_k'} t$ for all $k \in \set {1, \ldots, n}$.
By [[Definition:Smooth Path (Complex Analysis)|definition of smooth path]], it follows that $\gamma_k$ and $\gamma_k'$ are [[Definition:Continuous Complex Function|continuo... | Contour Integral is Well-Defined | https://proofwiki.org/wiki/Contour_Integral_is_Well-Defined | https://proofwiki.org/wiki/Contour_Integral_is_Well-Defined | [
"Complex Contour Integrals",
"Examples of Well-Defined Mappings"
] | [
"Definition:Contour/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"Definition:Complex Number/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Complex",
"Definition:Continuous Complex Function",
"Definition:Co... | [
"Definition:Smooth Path/Complex",
"Definition:Continuous Complex Function",
"Composite of Continuous Mappings is Continuous/Corollary",
"Combination Theorem for Continuous Functions/Complex/Sum Rule",
"Definition:Continuous Complex Function",
"Continuous Complex Function is Complex Riemann Integrable",
... |
proofwiki-6529 | Boolean Prime Ideal Theorem | Let $\struct {S, \le}$ be a Boolean lattice.
Let $I$ be an ideal in $S$.
Let $F$ be a filter on $S$.
Let $I \cap F = \O$.
Then there exists a prime ideal $P$ in $S$ such that:
:$I \subseteq P$
and:
:$P \cap F = \O$ | Let $Q$ be the set of all ideals of $S$ that are disjoint from $F$.
For each $x \in S$, define:
:$C_x = \set {q \in Q : x \in q}$
We want to construct a filter $\FF$ on $Q$ such that:
* $C_x \in \FF$ {{iff}} $x \in I$
* When the filter is extended to an ultrafilter $\UU$ by the Ultrafilter Lemma, the resulting structur... | Let $\struct {S, \le}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $S$.
Let $F$ be a [[Definition:Filter|filter]] on $S$.
Let $I \cap F = \O$.
Then there exists a [[Definition:Prime Ideal (Order Theory)|prime ideal]] $P$ in $S$ such that:
:$I \sub... | Let $Q$ be the [[Definition:Set|set]] of all [[Definition:Ideal (Order Theory)|ideals]] of $S$ that are [[Definition:Disjoint|disjoint]] from $F$.
For each $x \in S$, define:
:$C_x = \set {q \in Q : x \in q}$
We want to construct a [[Definition:Filter on Set|filter]] $\FF$ on $Q$ such that:
* $C_x \in \FF$ {{iff}} $... | Boolean Prime Ideal Theorem/Proof 2 | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem/Proof_2 | [
"Boolean Prime Ideal Theorem",
"Boolean Algebras"
] | [
"Definition:Boolean Lattice",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Prime Ideal (Order Theory)"
] | [
"Definition:Set",
"Definition:Ideal (Order Theory)",
"Definition:Disjoint",
"Definition:Filter on Set",
"Definition:Filter on Set",
"Definition:Ultrafilter on Set",
"Ultrafilter Lemma",
"Definition:Relative Complement",
"Axiom:Ideal Axioms (Order Theory)",
"Definition:Join (Order Theory)",
"Idea... |
proofwiki-6530 | Boolean Prime Ideal Theorem | Let $\struct {S, \le}$ be a Boolean lattice.
Let $I$ be an ideal in $S$.
Let $F$ be a filter on $S$.
Let $I \cap F = \O$.
Then there exists a prime ideal $P$ in $S$ such that:
:$I \subseteq P$
and:
:$P \cap F = \O$ | We prove that the Boolean Prime Ideal Theorem is equivalent to Stone's Representation Theorem for Boolean Algebras in ZF.
{{ProofWanted}} | Let $\struct {S, \le}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $S$.
Let $F$ be a [[Definition:Filter|filter]] on $S$.
Let $I \cap F = \O$.
Then there exists a [[Definition:Prime Ideal (Order Theory)|prime ideal]] $P$ in $S$ such that:
:$I \sub... | We prove that the [[Boolean Prime Ideal Theorem]] is equivalent to [[Stone's Representation Theorem for Boolean Algebras]] in ZF.
{{ProofWanted}} | Boolean Prime Ideal Theorem/Proof 3 | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem | https://proofwiki.org/wiki/Boolean_Prime_Ideal_Theorem/Proof_3 | [
"Boolean Prime Ideal Theorem",
"Boolean Algebras"
] | [
"Definition:Boolean Lattice",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Prime Ideal (Order Theory)"
] | [
"Boolean Prime Ideal Theorem",
"Stone's Representation Theorem for Boolean Algebras"
] |
proofwiki-6531 | Central Limit Theorem | Let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed real-valued random variables with:
:expectation $\expect {X_i} = \mu \in \R$
:variance $\var {X_i} = \sigma^2 > 0$
Let:
:$\ds S_n = \sum_{i \mathop = 1}^n X_i$
Then:
:$\dfrac {S_n - n \mu} {\sqrt {n \sigma^2} } \xrightarrow D \Gaussian 0 1$... | Let $Y_i = \dfrac {X_i - \mu} \sigma$.
We have that:
:$\expect {Y_i} = 0$
and:
:$\expect {Y_i^2} = 1$
Then by Taylor's Theorem the characteristic function can be written:
:$\map {\phi_{Y_i} } t = 1 - \dfrac {t^2} 2 + \map o {t^2}$
Now let:
{{begin-eqn}}
{{eqn | l = U_n
| r = \frac {S_n - n \mu} {\sqrt {n \sigma^... | Let $X_1, X_2, \ldots$ be a [[Definition:Sequence|sequence]] of [[Definition:Independent and Identically Distributed|independent and identically distributed]] [[Definition:Real-Valued Random Variable|real-valued random variables]] with:
:[[Definition:Expectation|expectation]] $\expect {X_i} = \mu \in \R$
:[[Definition... | Let $Y_i = \dfrac {X_i - \mu} \sigma$.
We have that:
:$\expect {Y_i} = 0$
and:
:$\expect {Y_i^2} = 1$
Then by [[Taylor's Theorem]] the [[Definition:Characteristic Function of Random Variable|characteristic function]] can be written:
:$\map {\phi_{Y_i} } t = 1 - \dfrac {t^2} 2 + \map o {t^2}$
Now let:
{{begin-eqn... | Central Limit Theorem | https://proofwiki.org/wiki/Central_Limit_Theorem | https://proofwiki.org/wiki/Central_Limit_Theorem | [
"Central Limit Theorem",
"Normal Distribution",
"Probability Theory",
"Named Theorems"
] | [
"Definition:Sequence",
"Definition:Random Sample (Probability Theory)",
"Definition:Random Variable/Real-Valued",
"Definition:Expectation",
"Definition:Variance",
"Definition:Convergence in Distribution",
"Definition:Standard Normal Distribution"
] | [
"Taylor's Theorem",
"Definition:Characteristic Function of Random Variable",
"Definition:Characteristic Function of Random Variable",
"Definition:Random Sample (Probability Theory)",
"Definition:Random Sample (Probability Theory)",
"Characteristic Function of Normal Distribution",
"Definition:Characteri... |
proofwiki-6532 | Arcsine Logarithmic Formulation | For any real number $x: -1 \le x \le 1$:
:$\arcsin x = \dfrac 1 i \map \ln {i x + \sqrt {1 - x^2} }$
where $\arcsin x$ is the arcsine and $i^2 = -1$. | Assume $y \in \R$ where $-\dfrac \pi 2 \le y \le \dfrac \pi 2$.
{{begin-eqn}}
{{eqn | l = y
| r = \arcsin x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sin y
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \frac 1 {2 i} \paren {e^{i y} - e^{-i y} }
|... | For any [[Definition:Real Number|real number]] $x: -1 \le x \le 1$:
:$\arcsin x = \dfrac 1 i \map \ln {i x + \sqrt {1 - x^2} }$
where $\arcsin x$ is the [[Definition:Real Arcsine|arcsine]] and $i^2 = -1$. | Assume $y \in \R$ where $-\dfrac \pi 2 \le y \le \dfrac \pi 2$.
{{begin-eqn}}
{{eqn | l = y
| r = \arcsin x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sin y
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \frac 1 {2 i} \paren {e^{i y} - e^{-i y} }
... | Arcsine Logarithmic Formulation | https://proofwiki.org/wiki/Arcsine_Logarithmic_Formulation | https://proofwiki.org/wiki/Arcsine_Logarithmic_Formulation | [
"Arcsine Function"
] | [
"Definition:Real Number",
"Definition:Inverse Sine/Real/Arcsine"
] | [
"Euler's Sine Identity",
"Square of Difference"
] |
proofwiki-6533 | Arccosine Logarithmic Formulation | For any real number $x: -1 \le x \le 1$:
:$\arccos x = \dfrac 1 i \map \ln {x + \sqrt {x^2 - 1} }$
where $\arccos x$ is the arccosine and $i^2 = -1$. | Assume $y \in \R$ such that $0 \le y \le \pi$.
{{begin-eqn}}
{{eqn | l = y
| r = \arccos x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cos y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \frac 1 2 \paren {e^{-i y} + e^{i y} }
| c = Euler's Cosine Identity
}}
{{eqn | ll= \leadstoandf... | For any [[Definition:Real Number|real number]] $x: -1 \le x \le 1$:
:$\arccos x = \dfrac 1 i \map \ln {x + \sqrt {x^2 - 1} }$
where $\arccos x$ is the [[Definition:Real Arccosine|arccosine]] and $i^2 = -1$. | Assume $y \in \R$ such that $0 \le y \le \pi$.
{{begin-eqn}}
{{eqn | l = y
| r = \arccos x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cos y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \frac 1 2 \paren {e^{-i y} + e^{i y} }
| c = [[Euler's Cosine Identity]]
}}
{{eqn | ll= \leadst... | Arccosine Logarithmic Formulation | https://proofwiki.org/wiki/Arccosine_Logarithmic_Formulation | https://proofwiki.org/wiki/Arccosine_Logarithmic_Formulation | [
"Arccosine Function"
] | [
"Definition:Real Number",
"Definition:Inverse Cosine/Real/Arccosine"
] | [
"Euler's Cosine Identity"
] |
proofwiki-6534 | Arctangent Logarithmic Formulation | For any real number $x$:
:$\arctan x = -\dfrac i 2 \map \ln {\dfrac {1 + i x} {1 - i x} }$
where $\arctan x$ is the arctangent and $i^2 = -1$. | Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
{{begin-eqn}}
{{eqn | l = y
| r = \arctan x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \tan y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = i \frac {1 - e^{2 i y} } {1 + e^{2 i y} }
| c = Euler's Tangent Identity
}}
{{eqn ... | For any [[Definition:Real Number|real number]] $x$:
:$\arctan x = -\dfrac i 2 \map \ln {\dfrac {1 + i x} {1 - i x} }$
where $\arctan x$ is the [[Definition:Real Arctangent|arctangent]] and $i^2 = -1$. | Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
{{begin-eqn}}
{{eqn | l = y
| r = \arctan x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \tan y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = i \frac {1 - e^{2 i y} } {1 + e^{2 i y} }
| c = [[Euler's Tangent Identity]]
}}
{... | Arctangent Logarithmic Formulation | https://proofwiki.org/wiki/Arctangent_Logarithmic_Formulation | https://proofwiki.org/wiki/Arctangent_Logarithmic_Formulation | [
"Arctangent Function"
] | [
"Definition:Real Number",
"Definition:Inverse Tangent/Real/Arctangent"
] | [
"Euler's Tangent Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-6535 | Length of Contour is Well-Defined | Let $C_1, \ldots, C_n$ be directed smooth curves.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.
Suppose that $\sigma_k: \closedint {c_k} {d_k} \to \C$ is a reparameter... | From the definition of directed smooth curve, it follows that $\sigma_k = \gamma_k \circ \phi_k$ for all $k \in \set {1, \ldots, n}$.
Here, $\phi_k: \closedint {c_k} {d_k} \to \closedint {a_k} {b_k}$ is a bijective differentiable strictly increasing function.
For all $k \in \set {1, \ldots, n}$, $\gamma_k$ and $\sigm... | Let $C_1, \ldots, C_n$ be [[Definition:Directed Smooth Curve|directed smooth curves]].
Let $C_k$ be [[Definition:Parameterization of Directed Smooth Curve|parameterized]] by the [[Definition:Smooth Path (Complex Analysis)|smooth path]] $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let... | From the [[Definition:Directed Smooth Curve|definition of directed smooth curve]], it follows that $\sigma_k = \gamma_k \circ \phi_k$ for all $k \in \set {1, \ldots, n}$.
Here, $\phi_k: \closedint {c_k} {d_k} \to \closedint {a_k} {b_k}$ is a [[Definition:Bijection|bijective]] [[Definition:Differentiable on Interval|d... | Length of Contour is Well-Defined | https://proofwiki.org/wiki/Length_of_Contour_is_Well-Defined | https://proofwiki.org/wiki/Length_of_Contour_is_Well-Defined | [
"Complex Contour Integrals",
"Contour Integrals"
] | [
"Definition:Directed Smooth Curve",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Smooth Path/Complex",
"Definition:Contour/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Definite Integral"
] | [
"Definition:Directed Smooth Curve",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Continuous Complex Function",
"Complex Modulus Function is Continuous",
"Composite of Continuous Mappings is Continuous/Coro... |
proofwiki-6536 | Two Ring is Boolean Ring | Let $2$ be the two ring.
Then $2$ is a Boolean ring. | From Ring of Integers Modulo m is Ring, $2$ is a ring with unity.
Furthermore, the identities:
:$0 \cdot 0 = 0$
:$1 \cdot 1 = 1$
show that $2$ is also an idempotent ring.
Hence the result, by definition of Boolean ring.
{{qed}} | Let $2$ be the [[Definition:Two Ring|two ring]].
Then $2$ is a [[Definition:Boolean Ring|Boolean ring]]. | From [[Ring of Integers Modulo m is Ring]], $2$ is a [[Definition:Ring with Unity|ring with unity]].
Furthermore, the identities:
:$0 \cdot 0 = 0$
:$1 \cdot 1 = 1$
show that $2$ is also an [[Definition:Idempotent Ring|idempotent ring]].
Hence the result, by definition of [[Definition:Boolean Ring|Boolean ring]].
{... | Two Ring is Boolean Ring | https://proofwiki.org/wiki/Two_Ring_is_Boolean_Ring | https://proofwiki.org/wiki/Two_Ring_is_Boolean_Ring | [
"Boolean Rings"
] | [
"Definition:Two Ring",
"Definition:Boolean Ring"
] | [
"Ring of Integers Modulo m is Ring",
"Definition:Ring with Unity",
"Definition:Idempotent Ring",
"Definition:Boolean Ring"
] |
proofwiki-6537 | Idempotent Ring is Commutative | Let $\struct {R, +, \circ}$ be an idempotent ring.
Denote with $0_R$ the zero of $R$.
Then $\struct {R, +, \circ}$ is a commutative ring. | Let $x, y \in R$.
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = \paren {x + y}^2
| c = {{Defof|Idempotent Ring}}
}}
{{eqn | r = x^2 + x \circ y + y \circ x + y^2
| c = Binomial Theorem: Ring Theory
}}
{{eqn | r = x + x \circ y + y \circ x + y
| c = {{Defof|Idempotent Ring}}
}}
{{end-eqn}}
Subtract... | Let $\struct {R, +, \circ}$ be an [[Definition:Idempotent Ring|idempotent ring]].
Denote with $0_R$ the [[Definition:Ring Zero|zero]] of $R$.
Then $\struct {R, +, \circ}$ is a [[Definition:Commutative Ring|commutative ring]]. | Let $x, y \in R$.
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = \paren {x + y}^2
| c = {{Defof|Idempotent Ring}}
}}
{{eqn | r = x^2 + x \circ y + y \circ x + y^2
| c = [[Binomial Theorem/Ring Theory|Binomial Theorem: Ring Theory]]
}}
{{eqn | r = x + x \circ y + y \circ x + y
| c = {{Defof|Idempo... | Idempotent Ring is Commutative | https://proofwiki.org/wiki/Idempotent_Ring_is_Commutative | https://proofwiki.org/wiki/Idempotent_Ring_is_Commutative | [
"Idempotent Rings"
] | [
"Definition:Idempotent Ring",
"Definition:Ring Zero",
"Definition:Commutative Ring"
] | [
"Binomial Theorem/Ring Theory",
"Idempotent Ring has Characteristic Two",
"Definition:Commutative Ring"
] |
proofwiki-6538 | Idempotent Ring has Characteristic Two | Let $\struct {R, +, \circ}$ be an idempotent non-null ring.
Denote with $0_R$ the zero of $R$.
Then $\struct {R, +, \circ}$ has characteristic $2$. | Let $x \in R$.
Then:
{{begin-eqn}}
{{eqn | l = x + x
| r = \paren {x + x}^2
| c = {{Defof|Idempotent Ring}}
}}
{{eqn | r = \paren {x + x} \paren {x + x}
}}
{{eqn | r = x \paren {x + x} + x \paren {x + x}
| c = {{Ring-axiom|D}}
}}
{{eqn | r = x^2 + x^2 + x^2 + x^2
| c = {{Ring-axiom|D}} again
}}
... | Let $\struct {R, +, \circ}$ be an [[Definition:Idempotent Ring|idempotent]] [[Definition:Non-Null Ring|non-null ring]].
Denote with $0_R$ the [[Definition:Ring Zero|zero]] of $R$.
Then $\struct {R, +, \circ}$ has [[Definition:Characteristic of Ring|characteristic]] $2$. | Let $x \in R$.
Then:
{{begin-eqn}}
{{eqn | l = x + x
| r = \paren {x + x}^2
| c = {{Defof|Idempotent Ring}}
}}
{{eqn | r = \paren {x + x} \paren {x + x}
}}
{{eqn | r = x \paren {x + x} + x \paren {x + x}
| c = {{Ring-axiom|D}}
}}
{{eqn | r = x^2 + x^2 + x^2 + x^2
| c = {{Ring-axiom|D}} again
}... | Idempotent Ring has Characteristic Two | https://proofwiki.org/wiki/Idempotent_Ring_has_Characteristic_Two | https://proofwiki.org/wiki/Idempotent_Ring_has_Characteristic_Two | [
"Idempotent Rings"
] | [
"Definition:Idempotent Ring",
"Definition:Non-Null Ring",
"Definition:Ring Zero",
"Definition:Characteristic of Ring"
] | [
"Definition:Group",
"Cancellation Laws"
] |
proofwiki-6539 | Two-Valued Functions form Boolean Ring | Let $S$ be a set, and let $2$ be the two ring.
Let $2^S$ be the set of all $2$-valued functions on $S$.
Denote with $+$ and $\cdot$ the pointwise operations induced on $2^S$ by $+_2$ and $\times_2$, respectively.
Then $\struct {2^S, +, \cdot}$ is a Boolean ring. | By Structure Induced by Ring Operations is Ring, $\struct {2^S, +, \cdot}$ is a ring.
By Unity of Induced Structure, $\struct {2^S, +, \cdot}$ also has a unity.
By Induced Structure is Idempotent, $\cdot$ is an idempotent operation.
Hence $\struct {2^S, +, \cdot}$ is a Boolean ring.
{{qed}} | Let $S$ be a [[Definition:Set|set]], and let $2$ be the [[Definition:Two Ring|two ring]].
Let $2^S$ be the [[Definition:Set of All Mappings|set of all]] [[Definition:Two-Valued Function|$2$-valued functions]] on $S$.
Denote with $+$ and $\cdot$ the [[Definition:Pointwise Operation|pointwise operations induced]] on $2... | By [[Structure Induced by Ring Operations is Ring]], $\struct {2^S, +, \cdot}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
By [[Unity of Induced Structure]], $\struct {2^S, +, \cdot}$ also has a [[Definition:Unity of Ring|unity]].
By [[Induced Structure is Idempotent]], $\cdot$ is an [[Definition:Idempotent Ope... | Two-Valued Functions form Boolean Ring | https://proofwiki.org/wiki/Two-Valued_Functions_form_Boolean_Ring | https://proofwiki.org/wiki/Two-Valued_Functions_form_Boolean_Ring | [
"Boolean Rings"
] | [
"Definition:Set",
"Definition:Two Ring",
"Definition:Set of All Mappings",
"Definition:Two-Valued Function",
"Definition:Pointwise Operation",
"Definition:Boolean Ring"
] | [
"Structure Induced by Ring Operations is Ring",
"Definition:Ring (Abstract Algebra)",
"Unity of Induced Structure",
"Definition:Unity (Abstract Algebra)/Ring",
"Induced Structure is Idempotent",
"Definition:Idempotence/Operation",
"Definition:Boolean Ring"
] |
proofwiki-6540 | Complex Integration by Substitution | Let $\closedint a b$ be a closed real interval.
Let $\phi: \closedint a b \to \R$ be a real function which has a derivative on $\closedint a b$.
Let $f: A \to \C$ be a continuous complex function, where $A$ is a subset of the image of $\phi$.
If $\map \phi a \le \map \phi b$, then:
:$\ds \int_{\map \phi a}^{\map \phi b... | Let $\Re$ and $\Im$ denote real parts and imaginary parts respectively.
Let $\map \phi a \le \map \phi b$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{\map \phi a}^{\map \phi b} \map f t \rd t
| r = \int_{\map \phi a}^{\map \phi b} \map \Re {\map f t} \rd t + i \int_{\map \phi a}^{\map \phi b} \map \Im {\map f t} \rd t... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\phi: \closedint a b \to \R$ be a [[Definition:Real Function|real function]] which has a [[Definition:Derivative|derivative]] on $\closedint a b$.
Let $f: A \to \C$ be a [[Definition:Continuous Complex Function|continuous complex... | Let $\Re$ and $\Im$ denote [[Definition:Real Part|real parts]] and [[Definition:Imaginary Part|imaginary parts]] respectively.
Let $\map \phi a \le \map \phi b$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{\map \phi a}^{\map \phi b} \map f t \rd t
| r = \int_{\map \phi a}^{\map \phi b} \map \Re {\map f t} \rd t + i... | Complex Integration by Substitution | https://proofwiki.org/wiki/Complex_Integration_by_Substitution | https://proofwiki.org/wiki/Complex_Integration_by_Substitution | [
"Complex Contour Integrals",
"Integration by Substitution"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Derivative",
"Definition:Continuous Complex Function",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Integration by Substitution/Definite Integral",
"Multiplication of Real and Imaginary Parts",
"Integration by Substitution/Definite Integral",
"Multiplication of Real and Imaginary Parts"
] |
proofwiki-6541 | Continuous Injection of Interval is Strictly Monotone | Let $I$ be a real interval.
Let $f: I \to \R$ be an injective continuous real function.
Then $f$ is strictly monotone. | {{AimForCont}} $f$ is not strictly monotone.
That is, there exist $x, y, z \in I$ with $x < y < z$ such that either:
:$\map f x \le \map f y$ and $\map f y \ge \map f z$
or:
:$\map f x \ge \map f y$ and $\map f y \le \map f z$
Suppose $\map f x \le \map f y$ and $\map f y \ge \map f z$.
If $\map f x = \map f y$, or $\m... | Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be an [[Definition:Injection|injective]] [[Definition:Continuous on Interval|continuous real function]].
Then $f$ is [[Definition:Strictly Monotone Real Function|strictly monotone]]. | {{AimForCont}} $f$ is not [[Definition:Strictly Monotone Real Function|strictly monotone]].
That is, there exist $x, y, z \in I$ with $x < y < z$ such that either:
:$\map f x \le \map f y$ and $\map f y \ge \map f z$
or:
:$\map f x \ge \map f y$ and $\map f y \le \map f z$
Suppose $\map f x \le \map f y$ and $\map f... | Continuous Injection of Interval is Strictly Monotone | https://proofwiki.org/wiki/Continuous_Injection_of_Interval_is_Strictly_Monotone | https://proofwiki.org/wiki/Continuous_Injection_of_Interval_is_Strictly_Monotone | [
"Continuous Functions"
] | [
"Definition:Real Interval",
"Definition:Injection",
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Monotone/Real Function"
] | [
"Definition:Strictly Monotone/Real Function",
"Definition:Injection",
"Definition:Contradiction",
"Definition:Continuous Real Function/Interval",
"Intermediate Value Theorem",
"Definition:Injection",
"Definition:Contradiction",
"Definition:Continuous Real Function/Interval",
"Intermediate Value Theo... |
proofwiki-6542 | Prime Ideal in Lattice | Let $\struct {L, \le}$ be a lattice.
Let $I$ be an ideal in $L$.
Then $I$ is a prime ideal {{iff}}:
:$\forall a, b \in L: a \wedge b \in I \implies a \in I \text{ or } b \in I$
where $a \wedge b$ denotes $\min \set {a, b}$, the meet of $a$ and $b$. | === Necessary Condition ===
Let $I$ be a prime ideal.
Let $a, b \in L$ such that $a, b \notin I$.
Then $a, b \in L \setminus I$.
By the definition of prime ideal $L \setminus I$ is a filter.
By the definition of a filter:
:$\exists c \in L \setminus I: c \le a, c \le b$
By the definition of meet:
:$c \le a \wedge b$
Si... | Let $\struct {L, \le}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Then $I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] {{iff}}:
:$\forall a, b \in L: a \wedge b \in I \implies a \in I \text{ or } b \in I$
where $a \wedge b$ denot... | === Necessary Condition ===
Let $I$ be a [[Definition:Prime Ideal (Order Theory)|prime ideal]].
Let $a, b \in L$ such that $a, b \notin I$.
Then $a, b \in L \setminus I$.
By the definition of [[Definition:Prime Ideal (Order Theory)|prime ideal]] $L \setminus I$ is a [[Definition:Filter|filter]].
By the definition ... | Prime Ideal in Lattice | https://proofwiki.org/wiki/Prime_Ideal_in_Lattice | https://proofwiki.org/wiki/Prime_Ideal_in_Lattice | [
"Order Theory"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Ideal (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Meet (Order Theory)"
] | [
"Definition:Prime Ideal (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Filter",
"Definition:Filter",
"Definition:Meet (Order Theory)",
"Rule of Transposition",
"Definition:Filter"
] |
proofwiki-6543 | Inverse of Increasing Bijection need not be Increasing | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a bijection which is increasing.
Then $\phi^{-1}: T \to S$ is not necessarily also increasing. | Proof by Counterexample:
Let $S := \powerset {\set {a, b} }$.
Let $T := \set {1, 2, 3, 4}$.
From Subset Relation on Power Set is Partial Ordering, $\struct {S, \subseteq}$ is an ordered set.
Clearly so is $\struct {T, \le}$ (although its ordering is in fact total, it is still technically an ordered set).
Let $\phi: S \... | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\phi: S \to T$ be a [[Definition:Bijection|bijection]] which is [[Definition:Increasing Mapping|increasing]].
Then $\phi^{-1}: T \to S$ is not necessarily also [[Definition:Increasing Mapping|increasing]]. | [[Proof by Counterexample]]:
Let $S := \powerset {\set {a, b} }$.
Let $T := \set {1, 2, 3, 4}$.
From [[Subset Relation on Power Set is Partial Ordering]], $\struct {S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
Clearly so is $\struct {T, \le}$ (although its [[Definition:Ordering|ordering]] is in fact... | Inverse of Increasing Bijection need not be Increasing | https://proofwiki.org/wiki/Inverse_of_Increasing_Bijection_need_not_be_Increasing | https://proofwiki.org/wiki/Inverse_of_Increasing_Bijection_need_not_be_Increasing | [
"Increasing Mappings",
"Bijections"
] | [
"Definition:Ordered Set",
"Definition:Bijection",
"Definition:Increasing/Mapping",
"Definition:Increasing/Mapping"
] | [
"Proof by Counterexample",
"Subset Relation on Power Set is Partial Ordering",
"Definition:Ordered Set",
"Definition:Ordering",
"Definition:Total Ordering",
"Definition:Ordered Set",
"Definition:Bijection",
"Definition:Increasing/Mapping",
"Definition:Increasing/Mapping"
] |
proofwiki-6544 | Order Isomorphic Sets are Equivalent | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be order isomorphic.
Then $S$ and $T$ are equivalent. | By definition, an order isomorphism is a bijection $\phi$ such that:
:$\phi: S \to T$ is order-preserving
:$\phi^{-1}: T \to S$ is order-preserving.
So, by definition, there exists a bijection between $S$ and $T$.
The result follows by definition of set equivalence.
{{qed}} | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be [[Definition:Order Isomorphism|order isomorphic]].
Then $S$ and $T$ are [[Definition:Set Equivalence|equivalent]]. | By definition, an [[Definition:Order Isomorphism|order isomorphism]] is a [[Definition:Bijection|bijection]] $\phi$ such that:
:$\phi: S \to T$ is [[Definition:Order-Preserving|order-preserving]]
:$\phi^{-1}: T \to S$ is [[Definition:Order-Preserving|order-preserving]].
So, by definition, there exists a [[Definition:... | Order Isomorphic Sets are Equivalent | https://proofwiki.org/wiki/Order_Isomorphic_Sets_are_Equivalent | https://proofwiki.org/wiki/Order_Isomorphic_Sets_are_Equivalent | [
"Examples of Equivalence Relations",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism",
"Definition:Set Equivalence"
] | [
"Definition:Order Isomorphism",
"Definition:Bijection",
"Definition:Increasing",
"Definition:Increasing",
"Definition:Bijection",
"Definition:Set Equivalence"
] |
proofwiki-6545 | Linear Combination of Contour Integrals | Let $C$ be a contour in $\C$.
Let $f, g: \Img C \to \C$ be continuous complex functions, where $\Img C$ denotes the image of $C$.
Let $\lambda, \mu \in \C$ be complex constants.
Then:
:$\ds \int_C \paren {\lambda \map f z + \mu \map g z} \rd z = \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z$ | By definition of contour, $C$ is a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_C \paren {\lambda \map f z + \mu \map g z} \rd z
| r = ... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour in $\C$]].
Let $f, g: \Img C \to \C$ be [[Definition:Continuous Complex Function|continuous complex functions]], where $\Img C$ denotes the [[Definition:Image of Contour (Complex Plane)|image]] of $C$.
Let $\lambda, \mu \in \C$ be [[Definition:Complex Number|c... | By definition of [[Definition:Contour (Complex Plane)|contour]], $C$ is a [[Definition:Finite Sequence|finite sequence]] $C_1, \ldots, C_n$ of [[Definition:Directed Smooth Curve|directed smooth curves]].
Let $C_k$ be [[Definition:Parameterization of Directed Smooth Curve (Complex Plane)|parameterized]] by the [[Defini... | Linear Combination of Contour Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Contour_Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Contour_Integrals | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Complex Plane",
"Definition:Continuous Complex Function",
"Definition:Contour/Image/Complex Plane",
"Definition:Complex Number",
"Definition:Constant"
] | [
"Definition:Contour/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Complex",
"Linear Combination of Complex Integrals"
] |
proofwiki-6546 | P-Product Metric Induces Product Topology | Let $M_A = \struct{A, d_A}$ and $M_B = \struct{B, d_B}$ be metric spaces.
Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively.
Let $p \ge 1$ be an extended real number.
Let $M = \struct{A \times B, d}$ be the $p$-product of $M_A$ and $M_B$.
We have that $M$ is a metric sp... | By $p$-Product Metrics are Lipschitz Equivalent and Lipschitz Equivalent Metrics are Topologically Equivalent, it suffices to consider the case $p = \infty$.
Let $\struct{A \times B, \tau'}$ be the product space of $\struct{A, \tau_A}$ and $\struct{B, \tau_B}$.
By the definition of $d$, it follows that an open ball in ... | Let $M_A = \struct{A, d_A}$ and $M_B = \struct{B, d_B}$ be [[Definition:Metric Space|metric spaces]].
Let $\tau_A$ and $\tau_B$ be the [[Definition:Topology Induced by Metric|topologies on $A$ and $B$ induced by $d_A$ and $d_B$]], respectively.
Let $p \ge 1$ be an [[Definition:Extended Real Number Line|extended real... | By [[P-Product Metrics are Lipschitz Equivalent|$p$-Product Metrics are Lipschitz Equivalent]] and [[Lipschitz Equivalent Metrics are Topologically Equivalent]], it suffices to consider the case $p = \infty$.
Let $\struct{A \times B, \tau'}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product sp... | P-Product Metric Induces Product Topology | https://proofwiki.org/wiki/P-Product_Metric_Induces_Product_Topology | https://proofwiki.org/wiki/P-Product_Metric_Induces_Product_Topology | [
"Metric Spaces",
"P-Product Metrics"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Extended Real Number Line",
"Definition:P-Product Metric",
"P-Product Metric is Metric",
"Definition:Topology Induced by Metric",
"Definition:Product Space (Topology)/Two Factor Spaces"
] | [
"P-Product Metrics are Lipschitz Equivalent",
"Lipschitz Equivalent Metrics are Topologically Equivalent",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Open Ball",
"Definition:Cartesian Product",
"Definition:Open Ball",
"Definition:Open Ball",
"Open Ball is Open Set/Pseudometri... |
proofwiki-6547 | P-Product Metrics are Lipschitz Equivalent | Let $M_A = \left({A, d_A}\right)$ and $M_B = \left({B, d_B}\right)$ be metric spaces.
Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively.
For all extended real numbers $p \ge 1$, let $M_p = \left({A \times B, d_p}\right)$ be the $p$-product of $M_A$ and $M_B$.
Then all t... | From $p$-Product Metric is Metric, $M_p$ is a metric space.
For all real numbers $p \ge 1$, it follows from the definition of $d_p$ that:
:$\forall x, y \in A \times B: d_{\infty} \left({x, y}\right) \le d_p \left({x, y}\right) \le 2^{1/p} d_{\infty} \left({x, y}\right)$
Hence, $d_p$ and $d_{\infty}$ are Lipschitz equi... | Let $M_A = \left({A, d_A}\right)$ and $M_B = \left({B, d_B}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $\tau_A$ and $\tau_B$ be the [[Definition:Topology Induced by Metric|topologies on $A$ and $B$ induced by $d_A$ and $d_B$]], respectively.
For all [[Definition:Extended Real Number Line|extended real... | From [[P-Product Metric is Metric|$p$-Product Metric is Metric]], $M_p$ is a [[Definition:Metric Space|metric space]].
For all [[Definition:Real Number|real numbers]] $p \ge 1$, it follows from the definition of $d_p$ that:
:$\forall x, y \in A \times B: d_{\infty} \left({x, y}\right) \le d_p \left({x, y}\right) \le ... | P-Product Metrics are Lipschitz Equivalent | https://proofwiki.org/wiki/P-Product_Metrics_are_Lipschitz_Equivalent | https://proofwiki.org/wiki/P-Product_Metrics_are_Lipschitz_Equivalent | [
"Lipschitz Equivalence",
"P-Product Metrics"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Extended Real Number Line",
"Definition:P-Product Metric",
"Definition:Metric Space/Metric",
"Definition:Lipschitz Equivalence/Metrics"
] | [
"P-Product Metric is Metric",
"Definition:Metric Space",
"Definition:Real Number",
"Definition:Lipschitz Equivalence/Metrics",
"Lipschitz Equivalence is Equivalence Relation",
"Category:Lipschitz Equivalence",
"Category:P-Product Metrics"
] |
proofwiki-6548 | Ordered Sum of Tosets is Totally Ordered Set/General Result | Let $S_1, S_2, \ldots, S_n$ all be tosets.
Let $T_n$ be the ordered sum of $S_1, S_2, \ldots, S_n$:
:$\forall n \in \N_{>0}: T_n = \begin{cases}
S_1 & : n = 1 \\
T_{n - 1} + S_n & : n > 1
\end{cases}$
Then $T_n$ is a toset. | From Ordered Sum of Tosets is Totally Ordered Set, $S_1 + S_2$ is a toset.
Suppose $T_{n-1}$ is a toset.
Given that $S_n$ is a toset, it follows from Ordered Sum of Tosets is Totally Ordered Set that $T_{n-1} + S_n$ is also a toset.
The result follows by induction.
{{qed}}
Category:Total Orderings
Category:Ordered Sums... | Let $S_1, S_2, \ldots, S_n$ all be [[Definition:Toset|tosets]].
Let $T_n$ be the [[Definition:Ordered Sum|ordered sum]] of $S_1, S_2, \ldots, S_n$:
:$\forall n \in \N_{>0}: T_n = \begin{cases}
S_1 & : n = 1 \\
T_{n - 1} + S_n & : n > 1
\end{cases}$
Then $T_n$ is a [[Definition:Toset|toset]]. | From [[Ordered Sum of Tosets is Totally Ordered Set]], $S_1 + S_2$ is a [[Definition:Toset|toset]].
Suppose $T_{n-1}$ is a [[Definition:Toset|toset]].
Given that $S_n$ is a [[Definition:Toset|toset]], it follows from [[Ordered Sum of Tosets is Totally Ordered Set]] that $T_{n-1} + S_n$ is also a [[Definition:Toset|to... | Ordered Sum of Tosets is Totally Ordered Set/General Result | https://proofwiki.org/wiki/Ordered_Sum_of_Tosets_is_Totally_Ordered_Set/General_Result | https://proofwiki.org/wiki/Ordered_Sum_of_Tosets_is_Totally_Ordered_Set/General_Result | [
"Total Orderings",
"Ordered Sums"
] | [
"Symbols:Abbreviations/T/Toset",
"Definition:Ordered Sum",
"Symbols:Abbreviations/T/Toset"
] | [
"Ordered Sum of Tosets is Totally Ordered Set",
"Symbols:Abbreviations/T/Toset",
"Symbols:Abbreviations/T/Toset",
"Symbols:Abbreviations/T/Toset",
"Ordered Sum of Tosets is Totally Ordered Set",
"Symbols:Abbreviations/T/Toset",
"Principle of Mathematical Induction",
"Category:Total Orderings",
"Cate... |
proofwiki-6549 | Supremum of Suprema | Let $\struct {S, \preceq}$ be an ordered set.
Let $\mathbb T \subseteq \powerset S$, where $\powerset S$ is the power set of $S$.
Suppose all $T \in \mathbb T$ admit a supremum $\sup T$ in $S$.
Then:
:$\sup \bigcup \mathbb T = \sup {\set {\sup T: T \in \mathbb T} }$
if one of these two quantities exists (in $S$). | Suppose that $s = \sup \bigcup \mathbb T \in S$.
By Set is Subset of Union, $T \subseteq \bigcup \mathbb T$ for all $T \in \mathbb T$.
Hence by Supremum of Subset:
:$\forall T \in \mathbb T: \sup T \preceq s$
Suppose now that $a \in S$ satisfies:
:$\forall T \in \mathbb T: \sup T \preceq a$
Then by transitivity of $\pr... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\mathbb T \subseteq \powerset S$, where $\powerset S$ is the [[Definition:Power Set|power set]] of $S$.
Suppose all $T \in \mathbb T$ admit a [[Definition:Supremum of Set|supremum]] $\sup T$ in $S$.
Then:
:$\sup \bigcup \mathbb T = \sup ... | Suppose that $s = \sup \bigcup \mathbb T \in S$.
By [[Set is Subset of Union]], $T \subseteq \bigcup \mathbb T$ for all $T \in \mathbb T$.
Hence by [[Supremum of Subset]]:
:$\forall T \in \mathbb T: \sup T \preceq s$
Suppose now that $a \in S$ satisfies:
:$\forall T \in \mathbb T: \sup T \preceq a$
Then by [[Def... | Supremum of Suprema | https://proofwiki.org/wiki/Supremum_of_Suprema | https://proofwiki.org/wiki/Supremum_of_Suprema | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Power Set",
"Definition:Supremum of Set"
] | [
"Set is Subset of Union",
"Supremum of Subset",
"Definition:Transitive Relation",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Transitive Relation",
"Definition:Supremum of Set",
"Category:Order Theory"
] |
proofwiki-6550 | Antilexicographic Product of Totally Ordered Sets is Totally Ordered/General Result | Let $S_1, S_2, \ldots, S_n$ all be totally ordered sets.
Let $T_n$ be the antilexicographic product of $S_1, S_2, \ldots, S_n$:
:$\forall n \in \N_{>0}: T_n = \begin {cases}
S_1 & : n = 1 \\
T_{n - 1} \otimes^a S_n & : n > 1
\end {cases}$
Then $T_n$ is a totally ordered set. | From Antilexicographic Product of Totally Ordered Sets is Totally Ordered, $S_1 \otimes^a S_2$ is a totally ordered set.
Suppose $T_{n - 1}$ is a totally ordered set.
Given that $S_n$ is a totally ordered set, it follows from Antilexicographic Product of Totally Ordered Sets is Totally Ordered that $T_{n - 1} \otimes^a... | Let $S_1, S_2, \ldots, S_n$ all be [[Definition:Totally Ordered Set|totally ordered sets]].
Let $T_n$ be the [[Definition:Antilexicographic Order|antilexicographic product]] of $S_1, S_2, \ldots, S_n$:
:$\forall n \in \N_{>0}: T_n = \begin {cases}
S_1 & : n = 1 \\
T_{n - 1} \otimes^a S_n & : n > 1
\end {cases}$
Then... | From [[Antilexicographic Product of Totally Ordered Sets is Totally Ordered]], $S_1 \otimes^a S_2$ is a [[Definition:Totally Ordered Set|totally ordered set]].
Suppose $T_{n - 1}$ is a [[Definition:Totally Ordered Set|totally ordered set]].
Given that $S_n$ is a [[Definition:Totally Ordered Set|totally ordered set]],... | Antilexicographic Product of Totally Ordered Sets is Totally Ordered/General Result | https://proofwiki.org/wiki/Antilexicographic_Product_of_Totally_Ordered_Sets_is_Totally_Ordered/General_Result | https://proofwiki.org/wiki/Antilexicographic_Product_of_Totally_Ordered_Sets_is_Totally_Ordered/General_Result | [
"Antilexicographic Product of Totally Ordered Sets is Totally Ordered"
] | [
"Definition:Totally Ordered Set",
"Definition:Antilexicographic Order",
"Definition:Totally Ordered Set"
] | [
"Antilexicographic Product of Totally Ordered Sets is Totally Ordered",
"Definition:Totally Ordered Set",
"Definition:Totally Ordered Set",
"Definition:Totally Ordered Set",
"Antilexicographic Product of Totally Ordered Sets is Totally Ordered",
"Definition:Totally Ordered Set",
"Principle of Mathematic... |
proofwiki-6551 | Derivative of Complex Composite Function | Let $f: D \to \C$ be a complex-differentiable function, where $D \subseteq \C$ is an open set.
Let $g: \Img f \to \C$ be a complex-differentiable function, where $\Img f$ denotes the image of $f$.
Define $h = f \circ g: D \to C$ as the composite of $f$ and $g$.
Then $h$ is complex-differentiable, and its derivative is ... | {{MissingLinks|among general reference to results, maybe also some links to details on this "method of variations" approach}}
Put $y = \map g z$.
Let $\delta z \in \C \setminus \set 0$.
Put $\delta y = \map g {z + \delta z} - y$, so:
:$\map g {z + \delta z} = y + \delta y$
As $\delta z \to 0$, we have:
:$(1): \quad \de... | Let $f: D \to \C$ be a [[Definition:Complex-Differentiable Function|complex-differentiable function]], where $D \subseteq \C$ is an [[Definition:Open Set (Complex Analysis)|open set]].
Let $g: \Img f \to \C$ be a [[Definition:Complex-Differentiable Function|complex-differentiable function]], where $\Img f$ denotes the... | {{MissingLinks|among general reference to results, maybe also some links to details on this "method of variations" approach}}
Put $y = \map g z$.
Let $\delta z \in \C \setminus \set 0$.
Put $\delta y = \map g {z + \delta z} - y$, so:
:$\map g {z + \delta z} = y + \delta y$
As $\delta z \to 0$, we have:
:$(1): \q... | Derivative of Complex Composite Function | https://proofwiki.org/wiki/Derivative_of_Complex_Composite_Function | https://proofwiki.org/wiki/Derivative_of_Complex_Composite_Function | [
"Complex Differential Calculus",
"Derivative of Composite Function"
] | [
"Definition:Differentiable Mapping/Complex Function",
"Definition:Open Set/Complex Analysis",
"Definition:Differentiable Mapping/Complex Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Composition of Mappings",
"Definition:Differentiable Mapping/Complex Function",
"Definition:Der... | [
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Definition:Differentiable Mapping/Complex Function"
] |
proofwiki-6552 | Ring of Idempotents is Idempotent Ring | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $\struct {A, \oplus, \circ}$ be its ring of idempotents.
Then $\struct {A, \oplus, \circ}$ is an idempotent ring. | First, it is to be established that $\struct {A, \oplus, \circ}$ is a ring in the first place.
This we do by verifying the ring axioms. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {A, \oplus, \circ}$ be its [[Definition:Ring of Idempotents|ring of idempotents]].
Then $\struct {A, \oplus, \circ}$ is an [[Definition:Idempotent Ring|idempotent ring]]. | First, it is to be established that $\struct {A, \oplus, \circ}$ is a [[Definition:Ring (Abstract Algebra)|ring]] in the first place.
This we do by verifying the [[Axiom:Ring Axioms|ring axioms]]. | Ring of Idempotents is Idempotent Ring | https://proofwiki.org/wiki/Ring_of_Idempotents_is_Idempotent_Ring | https://proofwiki.org/wiki/Ring_of_Idempotents_is_Idempotent_Ring | [
"Ring Theory",
"Idempotent Rings"
] | [
"Definition:Commutative Ring",
"Definition:Ring of Idempotents",
"Definition:Idempotent Ring"
] | [
"Definition:Ring (Abstract Algebra)",
"Axiom:Ring Axioms",
"Axiom:Ring Axioms",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-6553 | Reparameterization of Contour is Contour | Let $\closedint a b$ and $\closedint c d$ be closed real intervals.
Let $\gamma: \closedint a b \to \C$ be a contour in $\C$.
That is, there exists a subdivision $a_0, a_1 , \ldots, a_n$ of $\closedint a b$ such that $\gamma \restriction_{I_i}$ is a smooth path for all $i \in \set {1, \ldots, n}$, where $I_i = \closedi... | $\phi$ is either bijective or strictly increasing.
From Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone, it follows that in both cases $\phi$ is both bijective and strictly monotone.
As $\map \phi c < \map \phi d$, $\phi$ must be strictly increasing.
As $\phi$ is strictly increasing, we h... | Let $\closedint a b$ and $\closedint c d$ be [[Definition:Closed Real Interval|closed real intervals]].
Let $\gamma: \closedint a b \to \C$ be a [[Definition:Contour (Complex Plane)|contour]] in $\C$.
That is, there exists a [[Definition:Subdivision of Interval|subdivision]] $a_0, a_1 , \ldots, a_n$ of $\closedint a ... | $\phi$ is either [[Definition:Bijection|bijective]] or [[Definition:Strictly Increasing Real Function|strictly increasing]].
From [[Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone]], it follows that in both cases $\phi$ is both [[Definition:Bijection|bijective]] and [[Definition:Strictly... | Reparameterization of Contour is Contour | https://proofwiki.org/wiki/Reparameterization_of_Contour_is_Contour | https://proofwiki.org/wiki/Reparameterization_of_Contour_is_Contour | [
"Complex Contour Integrals"
] | [
"Definition:Real Interval/Closed",
"Definition:Contour/Complex Plane",
"Definition:Subdivision of Interval",
"Definition:Smooth Path/Complex",
"Definition:Restriction/Mapping",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Function",
"Definition:Bijection",
"Definition... | [
"Definition:Bijection",
"Definition:Strictly Increasing/Real Function",
"Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone",
"Definition:Bijection",
"Definition:Strictly Monotone/Real Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Increasing/... |
proofwiki-6554 | Contour Integral is Independent of Parameterization | Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.
S... | By definition of parameterization:
:$\gamma_k = \sigma_k \circ \phi_i$
for all $k \in \set {1, \ldots, n}$.
Here, $\phi_k: \closedint {c_k} {d_k} \to \closedint {a_k} {b_k}$ is a bijective differentiable strictly increasing real function.
Then:
{{begin-eqn}}
{{eqn | l = \int_C \map f z \rd z
| r = \sum_{k \matho... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour]] defined by a [[Definition:Finite Sequence|finite sequence]] $C_1, \ldots, C_n$ of [[Definition:Directed Smooth Curve|directed smooth curves]].
Let $C_k$ be [[Definition:Parameterization of Directed Smooth Curve|parameterized]] by the [[Definition:Smooth Path ... | By definition of [[Definition:Parameterization of Directed Smooth Curve|parameterization]]:
:$\gamma_k = \sigma_k \circ \phi_i$
for all $k \in \set {1, \ldots, n}$.
Here, $\phi_k: \closedint {c_k} {d_k} \to \closedint {a_k} {b_k}$ is a [[Definition:Bijection|bijective]] [[Definition:Differentiable on Interval|differ... | Contour Integral is Independent of Parameterization | https://proofwiki.org/wiki/Contour_Integral_is_Independent_of_Parameterization | https://proofwiki.org/wiki/Contour_Integral_is_Independent_of_Parameterization | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Smooth Path/Complex",
"Definition:Continuous Complex Function",
"Definition:Complex Function",
"Definition:Contour/Image/Complex Plane"... | [
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Complex Integration by Substitution",
"Derivative of Complex Composite Function",
"Reparameterization of Directed Smoo... |
proofwiki-6555 | Unity is Unity in Ring of Idempotents | Let $\left({R, +, \circ}\right)$ be a commutative and unitary ring whose unity is $1_R$.
Let $\left({A, \oplus, \circ}\right)$ be the ring of idempotents of $R$.
Then $1_R$ is also a unity for $\left({A, \oplus, \circ}\right)$. | From Unity of Ring is Idempotent, $1_R$ is an idempotent element of $R$.
Hence $1_R \in A$.
Recall that the ring product of $A$ is a restriction from that of $R$.
Hence, for each $x \in A$:
:$x \circ 1_R = x = 1_R \circ x$
so that $1_R$ is a unity for $A$, as desired.
{{qed}}
Category:Ring Theory
nspjhifbhflpwipy9lij3c... | Let $\left({R, +, \circ}\right)$ be a [[Definition:Commutative and Unitary Ring|commutative and unitary ring]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $\left({A, \oplus, \circ}\right)$ be the [[Definition:Ring of Idempotents|ring of idempotents]] of $R$.
Then $1_R$ is also a [[Definition:Unity of Ring... | From [[Unity of Ring is Idempotent]], $1_R$ is an [[Definition:Idempotent Element|idempotent element]] of $R$.
Hence $1_R \in A$.
Recall that the [[Definition:Ring Product|ring product]] of $A$ is a [[Definition:Restriction of Operation|restriction]] from that of $R$.
Hence, for each $x \in A$:
:$x \circ 1_R = x =... | Unity is Unity in Ring of Idempotents | https://proofwiki.org/wiki/Unity_is_Unity_in_Ring_of_Idempotents | https://proofwiki.org/wiki/Unity_is_Unity_in_Ring_of_Idempotents | [
"Ring Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Ring of Idempotents",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Unity of Ring is Idempotent",
"Definition:Idempotence/Element",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Restriction/Operation",
"Definition:Unity (Abstract Algebra)/Ring",
"Category:Ring Theory"
] |
proofwiki-6556 | Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring | Let $\struct {R, +, \circ}$ be a commutative and unitary ring.
Let $\struct {A, \oplus, \circ}$ be its ring of idempotents.
Then $\struct {A, \oplus, \circ}$ is a Boolean ring. | From Ring of Idempotents is Idempotent Ring, $\struct {A, \oplus, \circ}$ is an idempotent ring.
By Unity is Unity in Ring of Idempotents, $\struct {A, \oplus, \circ}$ is also a unitary ring.
Hence, by definition, $\struct {A, \oplus, \circ}$ is a Boolean ring.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative and unitary ring]].
Let $\struct {A, \oplus, \circ}$ be its [[Definition:Ring of Idempotents|ring of idempotents]].
Then $\struct {A, \oplus, \circ}$ is a [[Definition:Boolean Ring|Boolean ring]]. | From [[Ring of Idempotents is Idempotent Ring]], $\struct {A, \oplus, \circ}$ is an [[Definition:Idempotent Ring|idempotent ring]].
By [[Unity is Unity in Ring of Idempotents]], $\struct {A, \oplus, \circ}$ is also a [[Definition:Unitary Ring|unitary ring]].
Hence, by definition, $\struct {A, \oplus, \circ}$ is a [[... | Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring | https://proofwiki.org/wiki/Ring_of_Idempotents_of_Commutative_and_Unitary_Ring_is_Boolean_Ring | https://proofwiki.org/wiki/Ring_of_Idempotents_of_Commutative_and_Unitary_Ring_is_Boolean_Ring | [
"Ring Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring of Idempotents",
"Definition:Boolean Ring"
] | [
"Ring of Idempotents is Idempotent Ring",
"Definition:Idempotent Ring",
"Unity is Unity in Ring of Idempotents",
"Definition:Ring with Unity",
"Definition:Boolean Ring"
] |
proofwiki-6557 | Boolean Ring has Proper Zero Divisor | Let $\left({R, +, \circ}\right)$ be a Boolean ring whose zero is $0_R$.
Suppose that $R$ has more than two elements.
Then $R$ has a proper zero divisor. | Since $R$ has more than two elements, there exist distinct non-zero elements $x, y \in R$.
Note that $x + y \ne 0_R$ since $x$ and $y$ are distinct (by Idempotent Ring has Characteristic Two).
If $x \circ y = 0$, $x$ is a proper zero divisor.
If $x \circ y \ne 0$, then:
{{begin-eqn}}
{{eqn|l = \left({x \circ y}\right) ... | Let $\left({R, +, \circ}\right)$ be a [[Definition:Boolean Ring|Boolean ring]] whose [[Definition:Ring Zero|zero]] is $0_R$.
Suppose that $R$ has more than two elements.
Then $R$ has a [[Definition:Proper Zero Divisor|proper zero divisor]]. | Since $R$ has more than two elements, there exist distinct non-[[Definition:Ring Zero|zero]] elements $x, y \in R$.
Note that $x + y \ne 0_R$ since $x$ and $y$ are distinct (by [[Idempotent Ring has Characteristic Two]]).
If $x \circ y = 0$, $x$ is a [[Definition:Proper Zero Divisor|proper zero divisor]].
If $x \ci... | Boolean Ring has Proper Zero Divisor | https://proofwiki.org/wiki/Boolean_Ring_has_Proper_Zero_Divisor | https://proofwiki.org/wiki/Boolean_Ring_has_Proper_Zero_Divisor | [
"Boolean Rings"
] | [
"Definition:Boolean Ring",
"Definition:Ring Zero",
"Definition:Proper Zero Divisor"
] | [
"Definition:Ring Zero",
"Idempotent Ring has Characteristic Two",
"Definition:Proper Zero Divisor",
"Definition:Idempotent Ring",
"Idempotent Ring is Commutative",
"Idempotent Ring has Characteristic Two",
"Definition:Proper Zero Divisor",
"Proof by Cases"
] |
proofwiki-6558 | Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone | Let $\closedint a b$ and $\closedint c d$ be closed real intervals.
Let $f: \closedint c d \to \closedint a b$ be a continuous real function.
Let $\map f c, \map f d \in \set {a, b}$.
Then $f$ is bijective {{iff}} $f$ is strictly monotone. | === Necessary condition ===
Let $f$ be a bijection.
From Continuous Injection of Interval is Strictly Monotone, it follows that $f$ is strictly monotone.
{{qed|lemma}} | Let $\closedint a b$ and $\closedint c d$ be [[Definition:Closed Real Interval|closed real intervals]].
Let $f: \closedint c d \to \closedint a b$ be a [[Definition:Continuous Real Function|continuous real function]].
Let $\map f c, \map f d \in \set {a, b}$.
Then $f$ is [[Definition:Bijection|bijective]] {{iff}} $... | === Necessary condition ===
Let $f$ be a [[Definition:Bijection|bijection]].
From [[Continuous Injection of Interval is Strictly Monotone]], it follows that $f$ is [[Definition:Strictly Monotone Real Function|strictly monotone]].
{{qed|lemma}} | Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone | https://proofwiki.org/wiki/Continuous_Real_Function_on_Closed_Interval_is_Bijective_iff_Strictly_Monotone | https://proofwiki.org/wiki/Continuous_Real_Function_on_Closed_Interval_is_Bijective_iff_Strictly_Monotone | [
"Continuous Real Functions",
"Monotone Real Functions",
"Bijections"
] | [
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function",
"Definition:Bijection",
"Definition:Strictly Monotone/Real Function"
] | [
"Definition:Bijection",
"Continuous Injection of Interval is Strictly Monotone",
"Definition:Strictly Monotone/Real Function",
"Definition:Strictly Monotone/Real Function",
"Definition:Bijection"
] |
proofwiki-6559 | Concatenation of Contours is Contour | Let $C$ and $D$ be contours in the complex plane.
That is, $C$ is a finite sequence of directed smooth curves $C_1, \ldots, C_n$.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Similarly, $D$ is a finite sequence of directed smooth curves $D_... | By definition of contour, each $C_k$ and $D_j$ is a directed smooth curve for all $k \in \set {1, \ldots, n}, j \in \set {1, \ldots, m}$.
By definition of contour:
:$\map {\gamma_k} {b_k} = \map {\gamma_{k + 1} } {a_{k + 1} }$
and:
:$\map {\sigma_j} {d_j} = \map {\sigma_{j + 1} } {c_{j + 1} }$
for all $k \in \set {1, \... | Let $C$ and $D$ be [[Definition:Contour (Complex Plane)|contours]] in the [[Definition:Complex Plane|complex plane]].
That is, $C$ is a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Directed Smooth Curve|directed smooth curves]] $C_1, \ldots, C_n$.
Let $C_k$ be [[Definition:Parameterization of Direct... | By definition of [[Definition:Contour (Complex Plane)|contour]], each $C_k$ and $D_j$ is a [[Definition:Directed Smooth Curve|directed smooth curve]] for all $k \in \set {1, \ldots, n}, j \in \set {1, \ldots, m}$.
By definition of [[Definition:Contour (Complex Plane)|contour]]:
:$\map {\gamma_k} {b_k} = \map {\gamma_{... | Concatenation of Contours is Contour | https://proofwiki.org/wiki/Concatenation_of_Contours_is_Contour | https://proofwiki.org/wiki/Concatenation_of_Contours_is_Contour | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Smooth Path/Complex",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"D... | [
"Definition:Contour/Complex Plane",
"Definition:Directed Smooth Curve",
"Definition:Contour/Complex Plane",
"Definition:Contour/Complex Plane"
] |
proofwiki-6560 | Contour Integral of Concatenation of Contours | Let $C$ and $D$ be contours in $\C$.
Suppose that the end point of $C$ is equal to the start point of $D$, so the concatenation $C \cup D$ is defined.
Let $f: \Img {C \cup D} \to \C$ be a continuous complex function, where $\Img {C \cup D}$ denotes the image of $C \cup D$.
Then:
:$\ds \int \limits_{C \mathop \cup D} \m... | By definition of contour, $C$ is a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
Similarly, $D$ is a finite sequence $D_1, \ldots, D_m$ of directed smooth curves.
Let $D_j$ be par... | Let $C$ and $D$ be [[Definition:Contour (Complex Plane)|contours]] in $\C$.
Suppose that the [[Definition:End Point of Contour (Complex Plane)|end point]] of $C$ is equal to the [[Definition:Start Point of Contour (Complex Plane)|start point]] of $D$, so the [[Definition:Concatenation of Contours (Complex Plane)|conca... | By definition of [[Definition:Contour (Complex Plane)|contour]], $C$ is a [[Definition:Finite Sequence|finite sequence]] $C_1, \ldots, C_n$ of [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curves]].
Let $C_k$ be [[Definition:Parameterization of Directed Smooth Curve (Complex Plane)|parameterized]]... | Contour Integral of Concatenation of Contours | https://proofwiki.org/wiki/Contour_Integral_of_Concatenation_of_Contours | https://proofwiki.org/wiki/Contour_Integral_of_Concatenation_of_Contours | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Complex Plane",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Concatenation of Contours/Complex Plane",
"Definition:Continuous Complex Function",
"Definition:Complex Function",
"Definition:Contour/Image/Complex Plane"
] | [
"Definition:Contour/Complex Plane",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Complex",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve/Complex Plane",
"Defi... |
proofwiki-6561 | Regular Expression is Accepted by Finite State Machine | Let $R$ be a regular expression.
Then there exists a finite state machine $F$ such that its accepted language $\map L F$ is exactly $\map L R$, the language defined by $R$. | This proof proceeds by structural induction. | Let $R$ be a [[Definition:Regular Expression|regular expression]].
Then there exists a [[Definition:Finite State Machine|finite state machine]] $F$ such that its accepted language $\map L F$ is exactly $\map L R$, the language defined by $R$. | This proof proceeds by structural induction. | Regular Expression is Accepted by Finite State Machine | https://proofwiki.org/wiki/Regular_Expression_is_Accepted_by_Finite_State_Machine | https://proofwiki.org/wiki/Regular_Expression_is_Accepted_by_Finite_State_Machine | [
"Abstract Machines"
] | [
"Definition:Regular Expression",
"Definition:Finite State Machine"
] | [] |
proofwiki-6562 | Directed Smooth Curve Relation is Equivalence | Let $\sim$ denote a relation on the set of all smooth paths: $\set {\gamma: I \to \C : \text{$I$ is a closed real interval, $\gamma$ is a smooth path} }$.
Let $\gamma: \closedint a b \to \C$ and $\sigma : \closedint c d \to \C$ be smooth paths.
Define $\sim$ as follows:
:$\gamma \sim \sigma$ {{iff}} there exists a bij... | Checking in turn each of the criteria for an equivalence: | Let $\sim$ denote a [[Definition:Endorelation|relation]] on the [[Definition:Set|set]] of all [[Definition:Smooth Path (Complex Analysis)|smooth paths]]: $\set {\gamma: I \to \C : \text{$I$ is a closed real interval, $\gamma$ is a smooth path} }$.
Let $\gamma: \closedint a b \to \C$ and $\sigma : \closedint c d \to \C... | Checking in turn each of the criteria for an [[Definition:Equivalence Relation|equivalence]]: | Directed Smooth Curve Relation is Equivalence | https://proofwiki.org/wiki/Directed_Smooth_Curve_Relation_is_Equivalence | https://proofwiki.org/wiki/Directed_Smooth_Curve_Relation_is_Equivalence | [
"Smooth Paths (Complex Analysis)",
"Examples of Equivalence Relations"
] | [
"Definition:Endorelation",
"Definition:Set",
"Definition:Smooth Path/Complex",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Function",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-6563 | Reparameterization of Directed Smooth Curve Preserves Image | Let $\closedint a b$ and $\closedint c d$ be closed real intervals.
Let $\gamma : \closedint a b \to \C$ be a smooth path.
Let $C$ be a directed smooth curve with parameterization $\gamma$.
Suppose that $\sigma : \closedint c d \to \C$ is a reparameterization of $C$.
Then $\Img \gamma = \Img \sigma$, where $\Img \gamma... | By definition of directed smooth curve, there exists a bijective differentiable strictly increasing function $\phi: \closedint c d \to \closedint a b$ such that $\sigma = \gamma \circ \phi$.
From Surjection by Restriction of Codomain, it follows that there exists a function $\tilde \gamma: \closedint a b \to \Img \gamm... | Let $\closedint a b$ and $\closedint c d$ be [[Definition:Closed Real Interval|closed real intervals]].
Let $\gamma : \closedint a b \to \C$ be a [[Definition:Smooth Path (Complex Analysis)|smooth path]].
Let $C$ be a [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curve]] with [[Definition:Paramet... | By [[Definition:Directed Smooth Curve (Complex Plane)|definition of directed smooth curve]], there exists a [[Definition:Bijection|bijective]] [[Definition:Differentiable on Interval|differentiable]] [[Definition:Strictly Increasing Real Function|strictly increasing]] [[Definition:Real Function|function]] $\phi: \close... | Reparameterization of Directed Smooth Curve Preserves Image | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Preserves_Image | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Preserves_Image | [
"Directed Smooth Curves (Complex Plane)"
] | [
"Definition:Real Interval/Closed",
"Definition:Smooth Path/Complex",
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Function",
"Restriction of Mapping to Image is Surjection",
"Definition:Complex Function",
"Definition:... |
proofwiki-6564 | Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints | Let $\R^n$ be a real cartesian space of $n$ dimensions.
Let $\closedint a b$ and $\closedint c d$ be closed real intervals.
Let $\gamma: \closedint a b \to \R^n$ be a smooth path in $\R^n$.
Let $C$ be a directed smooth curve with parameterization $\gamma$.
Suppose that $\sigma: \closedint c d \to \R^n$ is a reparameter... | By definition of reparameterization, there exists a bijective differentiable strictly increasing real function $\phi: \closedint c d \closedint a b$ such that $\sigma = \gamma \circ \phi$.
As $\map {\phi^{-1} }{a} \in \closedint c d$:
: $c \le \map {\phi^{-1} }{a}$
As $\phi$ is strictly increasing:
: $\map \phi c \le \... | Let $\R^n$ be a [[Definition:Real Cartesian Space|real cartesian space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $\closedint a b$ and $\closedint c d$ be [[Definition:Closed Real Interval|closed real intervals]].
Let $\gamma: \closedint a b \to \R^n$ be a [[Definition:Smooth Path (Real Cartesi... | By definition of [[Definition:Parameterization of Directed Smooth Curve|reparameterization]], there exists a [[Definition:Bijection|bijective]] [[Definition:Differentiable on Interval|differentiable]] [[Definition:Strictly Increasing Real Function|strictly increasing]] [[Definition:Real Function|real function]] $\phi: ... | Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Maps_Endpoints_To_Endpoints | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Maps_Endpoints_To_Endpoints | [
"Vector Analysis"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Real Interval/Closed",
"Definition:Smooth Path/Real Cartesian Space",
"Definition:Directed Smooth Curve",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Directed Smo... | [
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Increasing/Real Functi... |
proofwiki-6565 | Product of Ring Negatives | Let $\struct {R, +, \circ}$ be a ring.
Then:
:$\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ \paren {-y} = x \circ y$
where $\paren {-x}$ denotes the negative of $x$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {-x} \circ \paren {-y}
| r = -\paren {x \circ \paren {-y} }
| c = Product with Ring Negative
}}
{{eqn | r = -\paren {-\paren {x \circ y} }
| c = Product with Ring Negative
}}
{{eqn | r = x \circ y
| c = Negative of Ring Negative
}}
{{end-eqn}}
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then:
:$\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ \paren {-y} = x \circ y$
where $\paren {-x}$ denotes the [[Definition:Ring Negative|negative]] of $x$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {-x} \circ \paren {-y}
| r = -\paren {x \circ \paren {-y} }
| c = [[Product with Ring Negative]]
}}
{{eqn | r = -\paren {-\paren {x \circ y} }
| c = [[Product with Ring Negative]]
}}
{{eqn | r = x \circ y
| c = [[Negative of Ring Negative]]
}}
{{end-eqn... | Product of Ring Negatives | https://proofwiki.org/wiki/Product_of_Ring_Negatives | https://proofwiki.org/wiki/Product_of_Ring_Negatives | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Negative"
] | [
"Product with Ring Negative",
"Product with Ring Negative",
"Negative of Ring Negative"
] |
proofwiki-6566 | Metric for Distinct Elements in Metric Space is Strictly Positive | Let $A$ be a set.
Let $d: A \times A \to \R$ be a real-valued function on $A$ with the following properties:
{{begin-axiom}}
{{axiom | n = \text M 1'
| q = \forall x, y \in A
| m = \map d {x, y} = 0 \iff x = y
}}
{{axiom | n = \text M 2
| q = \forall x, y, z \in A
| m = \map d {x, y} + \... | {{begin-eqn}}
{{eqn | q = \forall x, y \in A
| l = \map d {x, y} + \map d {y, x}
| o = \ge
| r = \map d {x, x}
| c = from {{Metric-space-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \map d {x, y}
| o = \ge
| r = 0
| c = from Axiom $(\text M 1')$ above and {{Metric-space-axio... | Let $A$ be a [[Definition:Set|set]].
Let $d: A \times A \to \R$ be a [[Definition:Real-Valued Function|real-valued function]] on $A$ with the following properties:
{{begin-axiom}}
{{axiom | n = \text M 1'
| q = \forall x, y \in A
| m = \map d {x, y} = 0 \iff x = y
}}
{{axiom | n = \text M 2
| ... | {{begin-eqn}}
{{eqn | q = \forall x, y \in A
| l = \map d {x, y} + \map d {y, x}
| o = \ge
| r = \map d {x, x}
| c = from {{Metric-space-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \map d {x, y}
| o = \ge
| r = 0
| c = from Axiom $(\text M 1')$ above and {{Metric-space-axio... | Metric for Distinct Elements in Metric Space is Strictly Positive | https://proofwiki.org/wiki/Metric_for_Distinct_Elements_in_Metric_Space_is_Strictly_Positive | https://proofwiki.org/wiki/Metric_for_Distinct_Elements_in_Metric_Space_is_Strictly_Positive | [
"Metrics",
"Metric Spaces"
] | [
"Definition:Set",
"Definition:Real-Valued Function",
"Axiom:Metric Space Axioms",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [
"Category:Metrics",
"Category:Metric Spaces"
] |
proofwiki-6567 | Transpose of Upper Triangular Matrix is Lower Triangular | The transpose of an upper triangular matrix is a lower triangular matrix. | Let $\mathbf U = \sqbrk a_{m n}$ be an upper triangular matrix.
By definition:
:$\forall a_{i j} \in \mathbf U: i > j \implies a_{i j} = 0$
Let $\mathbf U^\intercal = \sqbrk b_{n m}$ be the transpose of $\mathbf U$.
That is:
:$\mathbf U^\intercal = \sqbrk b_{n m}: \forall i \in \closedint 1 n, j \in \closedint 1 n: b_{... | The [[Definition:Transpose of Matrix|transpose]] of an [[Definition:Upper Triangular Matrix|upper triangular matrix]] is a [[Definition:Lower Triangular Matrix|lower triangular matrix]]. | Let $\mathbf U = \sqbrk a_{m n}$ be an [[Definition:Upper Triangular Matrix|upper triangular matrix]].
By definition:
:$\forall a_{i j} \in \mathbf U: i > j \implies a_{i j} = 0$
Let $\mathbf U^\intercal = \sqbrk b_{n m}$ be the [[Definition:Transpose of Matrix|transpose]] of $\mathbf U$.
That is:
:$\mathbf U^\inter... | Transpose of Upper Triangular Matrix is Lower Triangular | https://proofwiki.org/wiki/Transpose_of_Upper_Triangular_Matrix_is_Lower_Triangular | https://proofwiki.org/wiki/Transpose_of_Upper_Triangular_Matrix_is_Lower_Triangular | [
"Triangular Matrices",
"Upper Triangular Matrices",
"Lower Triangular Matrices",
"Transposes of Matrices"
] | [
"Definition:Transpose of Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix"
] | [
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Transpose of Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Category:Triangular Matrices",
"Category:Upper Triangular Matrices",
"Category:Lower Triangular Matrices",
"Category:Transposes of Matrices"
] |
proofwiki-6568 | Derivative iff Right and Left Derivative | Let $f$ be a real function.
Then $f$ is differentiable {{iff}} it has both right- and left-hand derivatives which agree. | Derivatives are defined in terms of limits.
The result follows from Limit iff Limits from Left and Right.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]].
Then $f$ is [[Definition:Differentiable Real Function|differentiable]] {{iff}} it has both [[Definition:Real Right-Hand Derivative|right-]] and [[Definition:Real Left-Hand Derivative|left-hand derivatives]] which [[Definition:Agreement of Mappings|agree]]. | [[Definition:Derivative of Real Function|Derivatives]] are defined in terms of [[Definition:Limit of Real Function|limits]].
The result follows from [[Limit iff Limits from Left and Right]].
{{qed}} | Derivative iff Right and Left Derivative | https://proofwiki.org/wiki/Derivative_iff_Right_and_Left_Derivative | https://proofwiki.org/wiki/Derivative_iff_Right_and_Left_Derivative | [
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Right-Hand Derivative/Real Function",
"Definition:Left-Hand Derivative/Real Function",
"Definition:Agreement/Mappings"
] | [
"Definition:Derivative/Real Function",
"Definition:Limit of Real Function",
"Limit iff Limits from Left and Right"
] |
proofwiki-6569 | Composition of Ring Homomorphisms is Ring Homomorphism | Let:
:$\struct {R_1, +_1, \odot_1}$
:$\struct {R_2, +_2, \odot_2}$
:$\struct {R_3, +_3, \odot_3}$
be rings.
Let:
:$\phi: \struct {R_1, +_1, \odot_1} \to \struct {R_2, +_2, \odot_2}$
:$\psi: \struct {R_2, +_2, \odot_2} \to \struct {R_3, +_3, \odot_3}$
be homomorphisms.
Then the composite of $\phi$ and $\psi$ is also a h... | A specific instance of Composite of Homomorphisms on Algebraic Structure is Homomorphism.
{{qed}} | Let:
:$\struct {R_1, +_1, \odot_1}$
:$\struct {R_2, +_2, \odot_2}$
:$\struct {R_3, +_3, \odot_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
:$\phi: \struct {R_1, +_1, \odot_1} \to \struct {R_2, +_2, \odot_2}$
:$\psi: \struct {R_2, +_2, \odot_2} \to \struct {R_3, +_3, \odot_3}$
be [[Definition:Ring Homomorp... | A specific instance of [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]].
{{qed}} | Composition of Ring Homomorphisms is Ring Homomorphism/Proof 1 | https://proofwiki.org/wiki/Composition_of_Ring_Homomorphisms_is_Ring_Homomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Homomorphisms_is_Ring_Homomorphism/Proof_1 | [
"Ring Homomorphisms",
"Composition of Ring Homomorphisms is Ring Homomorphism"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Homomorphism",
"Definition:Composition of Mappings",
"Definition:Ring Homomorphism"
] | [
"Composite of Homomorphisms is Homomorphism/Algebraic Structure"
] |
proofwiki-6570 | Composition of Ring Homomorphisms is Ring Homomorphism | Let:
:$\struct {R_1, +_1, \odot_1}$
:$\struct {R_2, +_2, \odot_2}$
:$\struct {R_3, +_3, \odot_3}$
be rings.
Let:
:$\phi: \struct {R_1, +_1, \odot_1} \to \struct {R_2, +_2, \odot_2}$
:$\psi: \struct {R_2, +_2, \odot_2} \to \struct {R_3, +_3, \odot_3}$
be homomorphisms.
Then the composite of $\phi$ and $\psi$ is also a h... | Let $\psi \circ \phi$ denote the composite of $\phi$ and $\psi$.
Then what we are trying to prove is denoted:
:$\paren {\psi \circ \phi}: \struct {R_1, +_1, \odot_1} \to \struct {R_3, +_3, \odot_3}$ is a homomorphism.
To prove the above is the case, we need to demonstrate that the morphism property is held by $+_1$ and... | Let:
:$\struct {R_1, +_1, \odot_1}$
:$\struct {R_2, +_2, \odot_2}$
:$\struct {R_3, +_3, \odot_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
:$\phi: \struct {R_1, +_1, \odot_1} \to \struct {R_2, +_2, \odot_2}$
:$\psi: \struct {R_2, +_2, \odot_2} \to \struct {R_3, +_3, \odot_3}$
be [[Definition:Ring Homomorp... | Let $\psi \circ \phi$ denote the [[Definition:Composition of Mappings|composite]] of $\phi$ and $\psi$.
Then what we are trying to prove is denoted:
:$\paren {\psi \circ \phi}: \struct {R_1, +_1, \odot_1} \to \struct {R_3, +_3, \odot_3}$ is a [[Definition:Ring Homomorphism|homomorphism]].
To prove the above is the ... | Composition of Ring Homomorphisms is Ring Homomorphism/Proof 2 | https://proofwiki.org/wiki/Composition_of_Ring_Homomorphisms_is_Ring_Homomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Homomorphisms_is_Ring_Homomorphism/Proof_2 | [
"Ring Homomorphisms",
"Composition of Ring Homomorphisms is Ring Homomorphism"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Homomorphism",
"Definition:Composition of Mappings",
"Definition:Ring Homomorphism"
] | [
"Definition:Composition of Mappings",
"Definition:Ring Homomorphism",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Group Homomorphism"
] |
proofwiki-6571 | Composition of Ring Epimorphisms is Ring Epimorphism | Let:
:$\struct {R_1, +_1, \circ_1}$
:$\struct {R_2, +_2, \circ_2}$
:$\struct {R_3, +_3, \circ_3}$
be rings.
Let:
:$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$
:$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$
be (ring) epimorphisms.
Then the composite of $\phi$ and $\psi$ is al... | A ring epimorphism is a ring homomorphism which is also a surection.
From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.
From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surection.
{{qed}} | Let:
:$\struct {R_1, +_1, \circ_1}$
:$\struct {R_2, +_2, \circ_2}$
:$\struct {R_3, +_3, \circ_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
:$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$
:$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$
be [[Definition:Ring Epimorp... | A [[Definition:Ring Epimorphism|ring epimorphism]] is a [[Definition:Ring Homomorphism|ring homomorphism]] which is also a [[Definition:Surjection|surection]].
From [[Composition of Ring Homomorphisms is Ring Homomorphism]], $\psi \circ \phi$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
From [[Composite ... | Composition of Ring Epimorphisms is Ring Epimorphism | https://proofwiki.org/wiki/Composition_of_Ring_Epimorphisms_is_Ring_Epimorphism | https://proofwiki.org/wiki/Composition_of_Ring_Epimorphisms_is_Ring_Epimorphism | [
"Ring Epimorphisms"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Epimorphism",
"Definition:Composition of Mappings",
"Definition:Ring Epimorphism"
] | [
"Definition:Ring Epimorphism",
"Definition:Ring Homomorphism",
"Definition:Surjection",
"Composition of Ring Homomorphisms is Ring Homomorphism",
"Definition:Ring Homomorphism",
"Composite of Surjections is Surjection",
"Definition:Surjection"
] |
proofwiki-6572 | Composition of Ring Monomorphisms is Ring Monomorphism | Let:
:$\struct {R_1, +_1, \circ_1}$
:$\struct {R_2, +_2, \circ_2}$
:$\struct {R_3, +_3, \circ_3}$
be rings.
Let:
:$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$
:$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$
be (ring) monomorphisms.
Then the composite of $\phi$ and $\psi$ is a... | A ring monomorphism is a ring homomorphism which is also an injection.
From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.
From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.
{{qed}} | Let:
:$\struct {R_1, +_1, \circ_1}$
:$\struct {R_2, +_2, \circ_2}$
:$\struct {R_3, +_3, \circ_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
:$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$
:$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$
be [[Definition:Ring Monomorp... | A [[Definition:Ring Monomorphism|ring monomorphism]] is a [[Definition:Ring Homomorphism|ring homomorphism]] which is also an [[Definition:Injection|injection]].
From [[Composition of Ring Homomorphisms is Ring Homomorphism]], $\psi \circ \phi$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
From [[Composit... | Composition of Ring Monomorphisms is Ring Monomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Monomorphisms_is_Ring_Monomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Monomorphisms_is_Ring_Monomorphism | [
"Ring Monomorphisms"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Monomorphism",
"Definition:Composition of Mappings",
"Definition:Ring Monomorphism"
] | [
"Definition:Ring Monomorphism",
"Definition:Ring Homomorphism",
"Definition:Injection",
"Composition of Ring Homomorphisms is Ring Homomorphism",
"Definition:Ring Homomorphism",
"Composite of Injections is Injection",
"Definition:Injection"
] |
proofwiki-6573 | Composition of Ring Isomorphisms is Ring Isomorphism | Let:
* $\left({R_1, +_1, \circ_1}\right)$
* $\left({R_2, +_2, \circ_2}\right)$
* $\left({R_3, +_3, \circ_3}\right)$
be rings.
Let:
* $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$
* $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$
be (ring) isomorphisms.
Then ... | A ring isomorphism is a ring homomorphism which is also a bijection.
From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.
From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.
{{qed}} | Let:
* $\left({R_1, +_1, \circ_1}\right)$
* $\left({R_2, +_2, \circ_2}\right)$
* $\left({R_3, +_3, \circ_3}\right)$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
* $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$
* $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_... | A [[Definition:Ring Isomorphism|ring isomorphism]] is a [[Definition:Ring Homomorphism|ring homomorphism]] which is also a [[Definition:Bijection|bijection]].
From [[Composition of Ring Homomorphisms is Ring Homomorphism]], $\psi \circ \phi$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
From [[Composite o... | Composition of Ring Isomorphisms is Ring Isomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Isomorphisms_is_Ring_Isomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Isomorphisms_is_Ring_Isomorphism | [
"Ring Isomorphisms"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Composition of Mappings",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Ring Homomorphism",
"Definition:Bijection",
"Composition of Ring Homomorphisms is Ring Homomorphism",
"Definition:Ring Homomorphism",
"Composite of Bijections is Bijection",
"Definition:Bijection"
] |
proofwiki-6574 | Composition of Ring Endomorphisms is Ring Endomorphism | Let $R$ be a set.
Let:
: $\struct {R, +_1, \circ_1}$
: $\struct {R, +_2, \circ_2}$
: $\struct {R, +_3, \circ_3}$
be rings.
Let:
: $\phi: \struct {R, +_1, \circ_1} \to \struct {R, +_2, \circ_2}$
: $\psi: \struct {R, +_2, \circ_2} \to \struct {R, +_3, \circ_3}$
be (ring) endomorphisms.
{{Questionable|They are between dif... | A ring endomorphism is a ring homomorphism $f$ from a set to itself.
That is:
:$\Dom \phi = \Cdm \phi$
:$\Dom \psi = \Cdm \psi$
From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.
By definition of composition of mappings:
:$\Cdm \phi = \Dom \psi$
Thus:
:$\Dom \phi = \C... | Let $R$ be a [[Definition:Set|set]].
Let:
: $\struct {R, +_1, \circ_1}$
: $\struct {R, +_2, \circ_2}$
: $\struct {R, +_3, \circ_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
: $\phi: \struct {R, +_1, \circ_1} \to \struct {R, +_2, \circ_2}$
: $\psi: \struct {R, +_2, \circ_2} \to \struct {R, +_3, \circ_3}$
... | A [[Definition:Ring Endomorphism|ring endomorphism]] is a [[Definition:Ring Homomorphism|ring homomorphism]] $f$ from a [[Definition:Set|set]] to itself.
That is:
:$\Dom \phi = \Cdm \phi$
:$\Dom \psi = \Cdm \psi$
From [[Composition of Ring Homomorphisms is Ring Homomorphism]], $\psi \circ \phi$ is a [[Definition:Rin... | Composition of Ring Endomorphisms is Ring Endomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Endomorphisms_is_Ring_Endomorphism | https://proofwiki.org/wiki/Composition_of_Ring_Endomorphisms_is_Ring_Endomorphism | [
"Ring Endomorphisms"
] | [
"Definition:Set",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Endomorphism",
"Definition:Underlying Set",
"Definition:Composition of Mappings",
"Definition:Ring Endomorphism"
] | [
"Definition:Ring Endomorphism",
"Definition:Ring Homomorphism",
"Definition:Set",
"Composition of Ring Homomorphisms is Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Composition of Mappings",
"Definition:Ring Endomorphism"
] |
proofwiki-6575 | Composition of Ring Automorphisms is Ring Automorphism | Let $R$ be a set.
Let:
* $\struct {R, +_1, \circ_1}$
* $\struct {R, +_2, \circ_2}$
* $\struct {R, +_3, \circ_3}$
be rings.
Let:
* $\phi: \struct {R, +_1, \circ_1} \to \struct {R, +_2, \circ_2}$
* $\psi: \struct {R, +_2, \circ_2} \to \struct {R, +_3, \circ_3}$
be ring automorphisms.
{{Questionable|They are between diffe... | A ring automorphism is a ring isomorphism $f$ from a set to itself.
That is:
:$\Dom \phi = \Cdm \phi$
:$\Dom \psi = \Cdm \psi$
From Composition of Ring Isomorphisms is Ring Isomorphism, $\psi \circ \phi$ is a ring isomorphism.
By definition of composition of mappings:
:$\Cdm \phi = \Dom \psi$
Thus:
:$\Dom \phi = \Cdm \... | Let $R$ be a [[Definition:Set|set]].
Let:
* $\struct {R, +_1, \circ_1}$
* $\struct {R, +_2, \circ_2}$
* $\struct {R, +_3, \circ_3}$
be [[Definition:Ring (Abstract Algebra)|rings]].
Let:
* $\phi: \struct {R, +_1, \circ_1} \to \struct {R, +_2, \circ_2}$
* $\psi: \struct {R, +_2, \circ_2} \to \struct {R, +_3, \circ_3}$
... | A [[Definition:Ring Automorphism|ring automorphism]] is a [[Definition:Ring Isomorphism|ring isomorphism]] $f$ from a [[Definition:Set|set]] to itself.
That is:
:$\Dom \phi = \Cdm \phi$
:$\Dom \psi = \Cdm \psi$
From [[Composition of Ring Isomorphisms is Ring Isomorphism]], $\psi \circ \phi$ is a [[Definition:Ring Is... | Composition of Ring Automorphisms is Ring Automorphism | https://proofwiki.org/wiki/Composition_of_Ring_Automorphisms_is_Ring_Automorphism | https://proofwiki.org/wiki/Composition_of_Ring_Automorphisms_is_Ring_Automorphism | [
"Ring Automorphisms",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Automorphism",
"Definition:Underlying Set",
"Definition:Composition of Mappings",
"Definition:Ring Automorphism"
] | [
"Definition:Ring Automorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Set",
"Composition of Ring Isomorphisms is Ring Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Composition of Mappings",
"Definition:Ring Automorphism"
] |
proofwiki-6576 | Set has Rank | Let $S$ be a set.
Then $S$ has a rank. | The proof shall proceed by Epsilon Induction on $S$.
Suppose that all the elements $a \in S$ have a rank.
That is, $a \in \map V x$ for some $x$.
Let:
:$\ds \map F a = \inf \set {x \in \On : a \in \map V x}$
be the rank of $a$.
Let:
:$\ds y = \sup \set {\map F a : a \in S}$
be the least level of the Von Neumann Hierarc... | Let $S$ be a [[Definition:Set|set]].
Then $S$ has a [[Definition:Rank (Set Theory)|rank]]. | The proof shall proceed by [[Epsilon Induction]] on $S$.
Suppose that all the elements $a \in S$ have a [[Definition:Rank (Set Theory)|rank]].
That is, $a \in \map V x$ for some $x$.
Let:
:$\ds \map F a = \inf \set {x \in \On : a \in \map V x}$
be the [[Definition:Rank (Set Theory)|rank]] of $a$.
Let:
:$\ds y = \s... | Set has Rank/Proof 1 | https://proofwiki.org/wiki/Set_has_Rank | https://proofwiki.org/wiki/Set_has_Rank/Proof_1 | [
"Set has Rank",
"Von Neumann Hierarchy"
] | [
"Definition:Set",
"Definition:Rank (Set Theory)"
] | [
"Epsilon Induction",
"Definition:Rank (Set Theory)",
"Definition:Rank (Set Theory)",
"Definition:Von Neumann Hierarchy",
"Definition:Element",
"Definition:Ordinal",
"Epsilon Induction",
"Definition:Rank (Set Theory)"
] |
proofwiki-6577 | Quotient Epimorphism is Epimorphism/Group | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $G / N$ be the quotient group of $G$ by $N$.
Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:
:$\forall x \in G: \map {q_N} x = x N$
Then $q_N$ is a group epimorphism whose kernel is $N$. | The proof follows from Quotient Mapping on Structure is Epimorphism.
When $N \lhd G$, we have:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \map {q_N} {x y}
| r = x y N
| c = {{Defof|Quotient Group Epimorphism}}
}}
{{eqn | r = \paren {x N} \paren {y N}
| c = {{Defof|Quotient Group}}
}}
{{e... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $G / N$ be the [[Definition:Quotient Group|quotient group]] of $G$ by $N$.
Let $q_N: G \to G / N$ be the [[Definition:Quotient Group Epimorphism|quotient epimorphism]] from $G$ to $G / N$:
:$\forall x \i... | The proof follows from [[Quotient Mapping on Structure is Epimorphism]].
When $N \lhd G$, we have:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \map {q_N} {x y}
| r = x y N
| c = {{Defof|Quotient Group Epimorphism}}
}}
{{eqn | r = \paren {x N} \paren {y N}
| c = {{Defof|Quotient Group}}
... | Quotient Epimorphism is Epimorphism/Group | https://proofwiki.org/wiki/Quotient_Epimorphism_is_Epimorphism/Group | https://proofwiki.org/wiki/Quotient_Epimorphism_is_Epimorphism/Group | [
"Quotient Epimorphism is Epimorphism",
"Quotient Groups",
"Group Epimorphisms",
"Quotient Epimorphisms"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Quotient Epimorphism/Group",
"Definition:Group Epimorphism",
"Definition:Kernel of Group Homomorphism"
] | [
"Quotient Mapping on Structure is Epimorphism",
"Definition:Group Homomorphism",
"Definition:Surjection",
"Definition:Group Epimorphism",
"Coset by Identity",
"Left Coset Equals Subgroup iff Element in Subgroup",
"Definition:Subset"
] |
proofwiki-6578 | Strictly Increasing Mapping on Well-Ordered Class | Let $\struct {S, \prec}$ be a strictly well-ordered class.
Let $\struct {T, <}$ be a strictly ordered class.
Let $f$ be a mapping from $S$ to $T$.
For each $i \in S$ such that $i$ is not maximal in $S$, let:
: $\map f i < \map f {\map \Succ i}$
where $\map \Succ i$ is the immediate successor element of $i$.
Let:
:$\for... | {{NotZFC}}
By Non-Greatest Element of Well-Ordered Class has Immediate Successor, $\map \Succ i$ is guaranteed to exist.
Let $i \prec j$.
Let $S_i := \set {q \in S: i \prec q}$.
{{explain|Important to specify what the domain of $q$ actually is -- presumably $S$.}}
Then $\map \Succ i$ is the minimal element of $S_i$.
By... | Let $\struct {S, \prec}$ be a strictly well-ordered class.
Let $\struct {T, <}$ be a strictly ordered class.
Let $f$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
For each $i \in S$ such that $i$ is not maximal in $S$, let:
: $\map f i < \map f {\map \Succ i}$
where $\map \Succ i$ is the [[Definition:Immediat... | {{NotZFC}}
By [[Non-Greatest Element of Well-Ordered Class has Immediate Successor]], $\map \Succ i$ is guaranteed to exist.
Let $i \prec j$.
Let $S_i := \set {q \in S: i \prec q}$.
{{explain|Important to specify what the domain of $q$ actually is -- presumably $S$.}}
Then $\map \Succ i$ is the minimal element of ... | Strictly Increasing Mapping on Well-Ordered Class | https://proofwiki.org/wiki/Strictly_Increasing_Mapping_on_Well-Ordered_Class | https://proofwiki.org/wiki/Strictly_Increasing_Mapping_on_Well-Ordered_Class | [
"Class Mappings",
"Increasing Mappings"
] | [
"Definition:Mapping",
"Definition:Immediate Successor Element"
] | [
"Non-Greatest Element of Well-Ordered Class has Immediate Successor",
"Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2",
"Category:Class Mappings",
"Category:Increasing Mappings"
] |
proofwiki-6579 | Non-Greatest Element of Well-Ordered Class has Immediate Successor | Let $C$ be a well-ordered class under an ordering $\le$.
Let $x \in C$.
Suppose that $x$ is not the greatest element in $C$.
Then $x$ has an immediate successor element in $C$. | Let $x$ be an element of $C$ which is not the greatest element of $C$.
Let $S$ be the class of successor elements of $x$ in $C$.
We have that $S$ is a subclass of $C$.
Also, $S$ is non-empty because $x$ is not the greatest element.
Thus $S$ is a non-empty subclass of $C$.
We have {{hypothesis}} that $\le$ is a well-ord... | Let $C$ be a [[Definition:Well-Ordered Class|well-ordered class]] under an [[Definition:Ordering (Class Theory)|ordering]] $\le$.
Let $x \in C$.
Suppose that $x$ is not the [[Definition:Greatest Element (Class Theory)|greatest element]] in $C$.
Then $x$ has an [[Definition:Immediate Successor Element|immediate succ... | Let $x$ be an [[Definition:Element of Class|element]] of $C$ which is not the [[Definition:Greatest Element (Class Theory)|greatest element]] of $C$.
Let $S$ be the [[Definition:Class (Class Theory)|class]] of [[Definition:Successor Element|successor elements]] of $x$ in $C$.
We have that $S$ is a [[Definition:Subcl... | Non-Greatest Element of Well-Ordered Class has Immediate Successor | https://proofwiki.org/wiki/Non-Greatest_Element_of_Well-Ordered_Class_has_Immediate_Successor | https://proofwiki.org/wiki/Non-Greatest_Element_of_Well-Ordered_Class_has_Immediate_Successor | [
"Well-Orderings",
"Successor Elements"
] | [
"Definition:Well-Ordered Class",
"Definition:Ordering/Class Theory",
"Definition:Greatest Element/Class Theory",
"Definition:Immediate Successor Element"
] | [
"Definition:Element/Class",
"Definition:Greatest Element/Class Theory",
"Definition:Class (Class Theory)",
"Definition:Succeed",
"Definition:Subclass",
"Definition:Non-Empty Set/Class Theory",
"Definition:Greatest Element/Class Theory",
"Definition:Non-Empty Set/Class Theory",
"Definition:Subclass",... |
proofwiki-6580 | Set Contained in Smallest Transitive Set | Let $S$ be a set.
Then there exists a transitive set $G$ such that:
:$S \subseteq G$
and:
:if $Q$ is any transitive set such that $S \subseteq Q$, then $G \subseteq Q$. | === Construction of $G$ ===
Let $U$ be the class of all sets.
{{explain|Perhaps Universal Class can be used here}}
Define the mapping $F: \N \to U$ recursively:
:$\map F 0 = S$
:$\map F {n + 1} = \bigcup \map F n$
Applying the axiom of union inductively, $\map F n$ is a set for each $n \in \N$.
Let $\ds G = \bigcup_{i ... | Let $S$ be a set.
Then there exists a transitive set $G$ such that:
:$S \subseteq G$
and:
:if $Q$ is any transitive set such that $S \subseteq Q$, then $G \subseteq Q$. | === Construction of $G$ ===
Let $U$ be the [[Definition:Class of all Sets|class of all sets]].
{{explain|Perhaps [[Definition:Universal Class|Universal Class]] can be used here}}
Define the [[Definition:Mapping|mapping]] $F: \N \to U$ recursively:
:$\map F 0 = S$
:$\map F {n + 1} = \bigcup \map F n$
Applying the a... | Set Contained in Smallest Transitive Set | https://proofwiki.org/wiki/Set_Contained_in_Smallest_Transitive_Set | https://proofwiki.org/wiki/Set_Contained_in_Smallest_Transitive_Set | [] | [] | [
"Definition:Class of all Sets",
"Definition:Universal Class",
"Definition:Mapping",
"Definition:Set"
] |
proofwiki-6581 | Transitive Set Contained in Von Neumann Hierarchy Level | Let $G$ be a transitive set.
Then for some ordinal $i$, $G \subseteq V_i$. | {{NotZFC}}
{{AimForCont}} for each ordinal $i$ the set $G \setminus V_i$ is non-empty.
Let $i$ be any ordinal.
Then by the axiom of foundation:
$\exists x: x \in G\setminus V_i \text{ and } x \cap \paren {G \setminus V_i} = \O$
Since $G$ is transitive, $x \subseteq G$.
Since $x \subseteq G$ and $x \cap \paren {G \setm... | Let $G$ be a [[Definition:Transitive Set|transitive set]].
Then for some [[Definition:Ordinal|ordinal]] $i$, $G \subseteq V_i$. | {{NotZFC}}
{{AimForCont}} for each ordinal $i$ the set $G \setminus V_i$ is non-empty.
Let $i$ be any ordinal.
Then by the [[Axiom:Axiom of Foundation|axiom of foundation]]:
$\exists x: x \in G\setminus V_i \text{ and } x \cap \paren {G \setminus V_i} = \O$
Since $G$ is transitive, $x \subseteq G$.
Since $x \sub... | Transitive Set Contained in Von Neumann Hierarchy Level | https://proofwiki.org/wiki/Transitive_Set_Contained_in_Von_Neumann_Hierarchy_Level | https://proofwiki.org/wiki/Transitive_Set_Contained_in_Von_Neumann_Hierarchy_Level | [
"Von Neumann Hierarchy"
] | [
"Definition:Transitive Class",
"Definition:Ordinal"
] | [
"Axiom:Axiom of Foundation",
"Strictly Increasing Mapping on Well-Ordered Class",
"Category:Von Neumann Hierarchy"
] |
proofwiki-6582 | External Direct Product Closure/General Result | Let $\ds \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ all be closed algebraic structures.... | Follows directly from External Direct Product Closure.
{{finish|Fill in the detail.}}
{{qed}}
Category:External Direct Product Closure
b3e434rvo0yrek6paldbzanjz0tu3f3 | Let $\ds \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the [[Definition:External Direct Product/General Definition|external direct product]] of the [[Definition:Algebraic Structure|algebraic structures]] $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Let $\struct {S_1, \circ_1... | Follows directly from [[External Direct Product Closure]].
{{finish|Fill in the detail.}}
{{qed}}
[[Category:External Direct Product Closure]]
b3e434rvo0yrek6paldbzanjz0tu3f3 | External Direct Product Closure/General Result | https://proofwiki.org/wiki/External_Direct_Product_Closure/General_Result | https://proofwiki.org/wiki/External_Direct_Product_Closure/General_Result | [
"External Direct Product Closure"
] | [
"Definition:External Direct Product/General Definition",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"External Direct Product Closure",
"Category:External Direct Product Closure"
] |
proofwiki-6583 | External Direct Product of Abelian Groups is Abelian Group/General Result | The external direct product of a finite sequence of abelian groups is itself an abelian group. | Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be abelian groups.
Let $\ds \struct {G, \circ} = \prod_{k \mathop = 1}^n G_k$ be the external direct product of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.
From External Direct Product of Groups is... | The [[Definition:External Direct Product|external direct product]] of a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Abelian Group|abelian groups]] is itself an [[Definition:Abelian Group|abelian group]]. | Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be [[Definition:Abelian Group|abelian groups]].
Let $\ds \struct {G, \circ} = \prod_{k \mathop = 1}^n G_k$ be the [[Definition:External Direct Product|external direct product]] of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldot... | External Direct Product of Abelian Groups is Abelian Group/General Result | https://proofwiki.org/wiki/External_Direct_Product_of_Abelian_Groups_is_Abelian_Group/General_Result | https://proofwiki.org/wiki/External_Direct_Product_of_Abelian_Groups_is_Abelian_Group/General_Result | [
"Abelian Groups",
"Group Direct Products"
] | [
"Definition:External Direct Product",
"Definition:Finite Sequence",
"Definition:Abelian Group",
"Definition:Abelian Group"
] | [
"Definition:Abelian Group",
"Definition:External Direct Product",
"External Direct Product of Groups is Group/Finite Product",
"Definition:Group",
"Definition:Commutative/Operation",
"External Direct Product Commutativity/General Result",
"Definition:Commutative/Operation",
"Definition:Abelian Group",... |
proofwiki-6584 | Homomorphism of External Direct Products/General Result | Let $n \in \N_{>0}$.
Let:
{{begin-eqn}}
{{eqn | l = \struct {\SS_n, \circledcirc_n}
| o = :=
| m = \prod_{k \mathop = 1}^n S_k
| mo= =
| r = \struct {S_1, \circ_1} \times \struct {S_2, \circ_2} \times \cdots \times \struct {S_n, \circ_n}
}}
{{eqn | l = \struct {\TT_n, \circledast_n}
| o = ... | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\Phi_n: \struct {\SS_n, \circledcirc_n} \to \struct {\TT_n, \circledast_n}$ is a homomorphism. | Let $n \in \N_{>0}$.
Let:
{{begin-eqn}}
{{eqn | l = \struct {\SS_n, \circledcirc_n}
| o = :=
| m = \prod_{k \mathop = 1}^n S_k
| mo= =
| r = \struct {S_1, \circ_1} \times \struct {S_2, \circ_2} \times \cdots \times \struct {S_n, \circ_n}
}}
{{eqn | l = \struct {\TT_n, \circledast_n}
| o =... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\Phi_n: \struct {\SS_n, \circledcirc_n} \to \struct {\TT_n, \circledast_n}$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]]. | Homomorphism of External Direct Products/General Result | https://proofwiki.org/wiki/Homomorphism_of_External_Direct_Products/General_Result | https://proofwiki.org/wiki/Homomorphism_of_External_Direct_Products/General_Result | [
"Homomorphism of External Direct Products"
] | [
"Definition:External Direct Product/General Definition",
"Definition:Algebraic Structure",
"Definition:Mapping",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Defini... |
proofwiki-6585 | Cesàro Mean | Let $\sequence {a_n}$ be a sequence of complex numbers.
Suppose that $\sequence {a_n}$ converges to $\ell$ in $\C$:
:$\ds \lim_{n \mathop \to \infty} a_n = \ell$
Then also:
:$\ds \lim_{n \mathop \to \infty} \frac {a_1 + \dotsb + a_n} n = \ell$ | For every fixed integer $n_0$, we write:
:$\ds \cmod {\frac {a_1 + \dotsb + a_n} n - \ell} \le \frac {\cmod {a_1 - \ell} + \dotsb + \cmod {a_n - \ell} } n \le \frac {n_0 \ds \sup_{k \mathop \le n_0} \cmod {a_k - \ell} } n + \sup_{n_0 \mathop < k \mathop \le n} \cmod {a_k - \ell}$
As $n$ tends to $+\infty$, we get:
:$\d... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Complex Number|complex numbers]].
Suppose that $\sequence {a_n}$ [[Definition:Convergent Sequence (Analysis)|converges]] to $\ell$ in $\C$:
:$\ds \lim_{n \mathop \to \infty} a_n = \ell$
Then also:
:$\ds \lim_{n \mathop \to \infty} \frac {a... | For every fixed integer $n_0$, we write:
:$\ds \cmod {\frac {a_1 + \dotsb + a_n} n - \ell} \le \frac {\cmod {a_1 - \ell} + \dotsb + \cmod {a_n - \ell} } n \le \frac {n_0 \ds \sup_{k \mathop \le n_0} \cmod {a_k - \ell} } n + \sup_{n_0 \mathop < k \mathop \le n} \cmod {a_k - \ell}$
As $n$ tends to $+\infty$, we get:
:... | Cesàro Mean | https://proofwiki.org/wiki/Cesàro_Mean | https://proofwiki.org/wiki/Cesàro_Mean | [
"Analysis",
"Topology",
"Limits of Sequences"
] | [
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Convergent Sequence/Analysis"
] | [] |
proofwiki-6586 | Characterization of Minimal Element | Let $C$ be a class.
Let $\prec$ be a relation on $C$.
Let $B$ be a subclass of $C$.
Let $x \in B$.
Let $S_x = \set {y \in C: y \prec x \text{ and } y \ne x}$ be the initial segment of $x$ in $C$.
Then $x$ is a minimal element of $B$ {{iff}} $B \cap S_x = \O$. | === Necessary Condition ===
Suppose $x$ is a minimal element of $B$.
Then for each $z \in B$ such that $z \ne x$, $z \nprec x$.
Thus $S_x \cap B = \O$.
{{qed|lemma}} | Let $C$ be a [[Definition:Class (Class Theory)|class]].
Let $\prec$ be a [[Definition:Relation|relation]] on $C$.
Let $B$ be a subclass of $C$.
Let $x \in B$.
Let $S_x = \set {y \in C: y \prec x \text{ and } y \ne x}$ be the initial segment of $x$ in $C$.
Then $x$ is a minimal element of $B$ {{iff}} $B \cap S_x = ... | === Necessary Condition ===
Suppose $x$ is a minimal element of $B$.
Then for each $z \in B$ such that $z \ne x$, $z \nprec x$.
Thus $S_x \cap B = \O$.
{{qed|lemma}} | Characterization of Minimal Element | https://proofwiki.org/wiki/Characterization_of_Minimal_Element | https://proofwiki.org/wiki/Characterization_of_Minimal_Element | [
"Class Theory"
] | [
"Definition:Class (Class Theory)",
"Definition:Relation"
] | [] |
proofwiki-6587 | Non-Empty Class has Element of Least Rank | Let $C$ be a class.
Let $C \ne \O$.
Then $C$ has an element of least rank.
That is:
:$\exists x \in C: \forall y \in C: \map {\operatorname {rank} } x \le \map {\operatorname {rank} } y$
where $\map {\operatorname {rank} } x$ is the rank of $x$. | {{NotZFC}}
By Set has Rank, each element of $C$ has a rank.
Let $R$ be the class of ranks of elements of $C$.
$R$ is non-empty because $C$ is non-empty.
{{explain|This should follow from arbitrary intersections of ordinals being ordinals.}}
Since any non-empty class of ordinals has a least element, $R$ has a least elem... | Let $C$ be a [[Definition:Class (Class Theory)|class]].
Let $C \ne \O$.
Then $C$ has an [[Definition:Element|element]] of least [[Definition:Rank (Set Theory)|rank]].
That is:
:$\exists x \in C: \forall y \in C: \map {\operatorname {rank} } x \le \map {\operatorname {rank} } y$
where $\map {\operatorname {rank} } x... | {{NotZFC}}
By [[Set has Rank]], each [[Definition:Element|element]] of $C$ has a [[Definition:Rank (Set Theory)|rank]].
Let $R$ be the [[Definition:Class (Class Theory)|class]] of [[Definition:Rank (Set Theory)|ranks]] of [[Definition:Element|element]]s of $C$.
$R$ is [[Definition:Non-Empty Class|non-empty]] because... | Non-Empty Class has Element of Least Rank | https://proofwiki.org/wiki/Non-Empty_Class_has_Element_of_Least_Rank | https://proofwiki.org/wiki/Non-Empty_Class_has_Element_of_Least_Rank | [
"Von Neumann Hierarchy"
] | [
"Definition:Class (Class Theory)",
"Definition:Element",
"Definition:Rank (Set Theory)",
"Definition:Rank (Set Theory)"
] | [
"Set has Rank",
"Definition:Element",
"Definition:Rank (Set Theory)",
"Definition:Class (Class Theory)",
"Definition:Rank (Set Theory)",
"Definition:Element",
"Definition:Non-Empty Set/Class Theory",
"Definition:Non-Empty Set/Class Theory",
"Definition:Non-Empty Set/Class Theory",
"Definition:Clas... |
proofwiki-6588 | Reversed Directed Smooth Curve is Directed Smooth Curve | Let $C$ be a directed smooth curve in $\C$.
Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$.
Define $\psi: \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ by $\psi \left({t}\right) = a + b - t$.
Define $\rho: \left[{a \,.\,.\, b}\right] \to \C$ by $\rho = \gamma \ci... | First, we prove that $\rho$ is a smooth path:
{{begin-eqn}}
{{eqn | l = \rho' \left({t}\right)
| r = \gamma' \left({\psi\left({t}\right) }\right) \psi' \left({t}\right)
| c = Derivative of Complex Composite Function
}}
{{eqn | r = -\gamma' \left({\psi\left({t}\right) }\right)
| c = Derivatives of Func... | Let $C$ be a [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curve]] in $\C$.
Let $C$ be [[Definition:Parameterization of Directed Smooth Curve (Complex Plane)|parameterized]] by the [[Definition:Smooth Path (Complex Analysis)|smooth path]] $\gamma: \left[{a \,.\,.\, b}\right] \to \C$.
Define $\ps... | First, we prove that $\rho$ is a [[Definition:Smooth Path (Complex Analysis)|smooth path]]:
{{begin-eqn}}
{{eqn | l = \rho' \left({t}\right)
| r = \gamma' \left({\psi\left({t}\right) }\right) \psi' \left({t}\right)
| c = [[Derivative of Complex Composite Function]]
}}
{{eqn | r = -\gamma' \left({\psi\left(... | Reversed Directed Smooth Curve is Directed Smooth Curve | https://proofwiki.org/wiki/Reversed_Directed_Smooth_Curve_is_Directed_Smooth_Curve | https://proofwiki.org/wiki/Reversed_Directed_Smooth_Curve_is_Directed_Smooth_Curve | [
"Directed Smooth Curves (Complex Plane)"
] | [
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Complex",
"Definition:Smooth Path/Complex",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Directed Smooth Curve/Complex Plane",
"D... | [
"Definition:Smooth Path/Complex",
"Derivative of Complex Composite Function",
"Derivatives of Function of a x + b",
"Definition:Continuously Differentiable",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Definition:Directed Smooth Curve/Parameterization/Complex Pla... |
proofwiki-6589 | Reversed Contour is Contour | Let $\R^n$ be a real cartesian space of $n$ dimensions.
Let $C$ be a contour in $\R^n$ that is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$.
Then the finite sequence of reversed directed smooth curves:
:$-C_n, -C_{n - 1}, \ldots, -C_1$
defines a contour that is ... | Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C_i$ is independent of the paraterization $\gamma_i$ of $C_i$.
We now prove that the end point of $-C_i$ is equal t... | Let $\R^n$ be a [[Definition:Real Cartesian Space|real cartesian space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $C$ be a [[Definition:Contour|contour in $\R^n$]] that is defined as a [[Definition:Concatenation of Contours|concatenation]] of a [[Definition:Finite Sequence|finite sequence]] $C_1... | Let $C_i$ be [[Definition:Parameterization of Directed Smooth Curve|parameterized]] by the [[Definition:Smooth Path (Complex Analysis)|smooth path]] $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
From [[Reversed Directed Smooth Curve is Directed Smooth Curve]], it follows that $-C_i$ is... | Reversed Contour is Contour | https://proofwiki.org/wiki/Reversed_Contour_is_Contour | https://proofwiki.org/wiki/Reversed_Contour_is_Contour | [
"Contour Integrals"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Contour",
"Definition:Concatenation of Contours",
"Definition:Finite Sequence",
"Definition:Directed Smooth Curve",
"Definition:Reversed Directed Smooth Curve",
"Definition:Contour"... | [
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Smooth Path/Complex",
"Reversed Directed Smooth Curve is Directed Smooth Curve",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Contour/Endpoints/Complex Plane",
"Definition:Reversed Directed Smooth Curve",
"Definition:Directed ... |
proofwiki-6590 | Contour Integral along Reversed Contour | Let $C$ be a contour in the complex plane $\C$.
Let $f: \Img C \to \C$ be a continuous complex functions, where $\Img C$ denotes the image of $C$.
Then the contour integral of $f$ along the reversed contour $-C$ is:
:$\ds \int_{-C} \map f z \rd z = -\int_C \map f z \rd z$ | First, suppose that $C$ is a directed smooth curve in $\C$.
Let $C$ be parameterized by the smooth path $\gamma: \closedint a b \to \C$.
By definition of reversed directed smooth curve, $-C$ is parameterized by a smooth path $\rho: \closedint a b \to \C$ with $\rho = \gamma \circ \psi$.
Here, $\psi: \closedint a b \to ... | Let $C$ be a [[Definition:Contour (Complex Plane)|contour]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Let $f: \Img C \to \C$ be a [[Definition:Continuous Complex Function|continuous]] [[Definition:Complex Function|complex functions]], where $\Img C$ denotes the [[Definition:Image of Contour (Complex Plan... | First, suppose that $C$ is a [[Definition:Directed Smooth Curve (Complex Plane)|directed smooth curve]] in $\C$.
Let $C$ be [[Definition:Parameterization of Directed Smooth Curve (Complex Plane)|parameterized]] by the [[Definition:Smooth Path (Complex Analysis)|smooth path]] $\gamma: \closedint a b \to \C$.
By defini... | Contour Integral along Reversed Contour | https://proofwiki.org/wiki/Contour_Integral_along_Reversed_Contour | https://proofwiki.org/wiki/Contour_Integral_along_Reversed_Contour | [
"Complex Contour Integrals"
] | [
"Definition:Contour/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Continuous Complex Function",
"Definition:Complex Function",
"Definition:Contour/Image/Complex Plane",
"Definition:Contour Integral/Complex",
"Definition:Reversed Contour/Complex Plane"
] | [
"Definition:Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Complex",
"Definition:Reversed Directed Smooth Curve/Complex Plane",
"Definition:Directed Smooth Curve/Parameterization/Complex Plane",
"Definition:Smooth Path/Compl... |
proofwiki-6591 | Strictly Well-Founded Relation determines Strictly Minimal Elements/Lemma | Let $A$ be a non-empty class.
Let $\RR$ be a strictly well-founded relation on $A$.
Then $A$ has a strictly minimal element under $\RR$. | {{NotZFC}}
The general strategy of the proof is as follows:
We will recursively define a certain subset, $a$, of $A$.
We will use the fact that $\RR$ is a strictly well-founded relation to choose a strictly minimal element $m$ of $a$.
Then we will prove that $m$ is in fact a strictly minimal element of $A$.
For each $x... | Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Class (Class Theory)|class]].
Let $\RR$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $A$.
Then $A$ has a [[Definition:Strictly Minimal Element|strictly minimal element]] under $\RR$. | {{NotZFC}}
The general strategy of the proof is as follows:
We will recursively define a certain [[Definition:Subset|subset]], $a$, of $A$.
We will use the fact that $\RR$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] to choose a [[Definition:Strictly Minimal Element|strictly mini... | Strictly Well-Founded Relation determines Strictly Minimal Elements/Lemma | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_determines_Strictly_Minimal_Elements/Lemma | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_determines_Strictly_Minimal_Elements/Lemma | [
"Class Theory"
] | [
"Definition:Non-Empty Set",
"Definition:Class (Class Theory)",
"Definition:Strictly Well-Founded Relation",
"Definition:Strictly Minimal Element"
] | [
"Definition:Subset",
"Definition:Strictly Well-Founded Relation",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Definition:Preimage/Relation/Element",
"Definition:Rank (Set Theory)",
"Definition:Smallest Element",
"Definition:Ordinal",
"Definition:Von Neumann Hierarc... |
proofwiki-6592 | Stone Space of Boolean Lattice is Topological Space | Let $\struct {B, \preceq, \wedge, \vee}$ be a Boolean lattice.
Let $\struct {U, \tau}$ be the Stone space of $B$.
Then $\struct {U, \tau}$ is a topological space. | The topology of the Stone space is defined as the topology generated by the basis $Q$ consisting of all sets of the form
:$\set {x \in U: b \in x}$
for some $b \in B$, where $U$ is the set of all ultrafilters on $B$.
By Union from Synthetic Basis is Topology, it suffices to show that $Q$ is a synthetic basis.
First, we... | Let $\struct {B, \preceq, \wedge, \vee}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $\struct {U, \tau}$ be the [[Definition:Stone Space/Boolean Lattice|Stone space]] of $B$.
Then $\struct {U, \tau}$ is a [[Definition:Topological Space|topological space]]. | The topology of the [[Definition:Stone Space/Boolean Lattice|Stone space]] is defined as the [[Definition:Topology Generated by Synthetic Basis|topology generated by the basis]] $Q$ consisting of all sets of the form
:$\set {x \in U: b \in x}$
for some $b \in B$, where $U$ is the [[Definition:Set|set]] of all [[Definit... | Stone Space of Boolean Lattice is Topological Space | https://proofwiki.org/wiki/Stone_Space_of_Boolean_Lattice_is_Topological_Space | https://proofwiki.org/wiki/Stone_Space_of_Boolean_Lattice_is_Topological_Space | [
"Stone Spaces"
] | [
"Definition:Boolean Lattice",
"Definition:Stone Space/Boolean Lattice",
"Definition:Topological Space"
] | [
"Definition:Stone Space/Boolean Lattice",
"Definition:Topology Generated by Synthetic Basis",
"Definition:Set",
"Definition:Ultrafilter (Order Theory)",
"Union from Synthetic Basis is Topology",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Cover of Set",
"Axiom:Filter Axioms",
"Definit... |
proofwiki-6593 | Ring Without Unity may have Quotient Ring with Unity | Let $\struct {R, +, \circ}$ be a ring.
Let $I$ be an ideal of $R$.
Let $\struct {R / I, +, \circ}$ be the associated quotient ring.
Then $\struct {R / I, +, \circ}$ may have a unity even if $\struct {R, +, \circ}$ has not. | Consider the external direct product of rings $\Z \oplus 2 \Z$.
From Integer Multiples form Commutative Ring, $2 \Z$ does not admit a unity.
By Unity of External Direct Sum of Rings, neither does $\Z \oplus 2 \Z$.
Now consider the ideal $\set 0 \times 2 \Z$ of $\Z \oplus 2 \Z$.
We have for all $a \in \Z$ and $b, c \in ... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $I$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $\struct {R / I, +, \circ}$ be the associated [[Definition:Quotient Ring|quotient ring]].
Then $\struct {R / I, +, \circ}$ may have a [[Definition:Unity of Ring|unity]] even if ... | Consider the [[Definition:Ring Direct Product|external direct product of rings]] $\Z \oplus 2 \Z$.
From [[Integer Multiples form Commutative Ring]], $2 \Z$ does not admit a [[Definition:Unity of Ring|unity]].
By [[Unity of External Direct Sum of Rings]], neither does $\Z \oplus 2 \Z$.
Now consider the [[Definition... | Ring Without Unity may have Quotient Ring with Unity | https://proofwiki.org/wiki/Ring_Without_Unity_may_have_Quotient_Ring_with_Unity | https://proofwiki.org/wiki/Ring_Without_Unity_may_have_Quotient_Ring_with_Unity | [
"Rings with Unity",
"Quotient Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Definition:Ring Direct Product",
"Integer Multiples form Commutative Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Unity of External Direct Sum of Rings",
"Definition:Ideal of Ring",
"Test for Ideal",
"Definition:Ideal of Ring",
"Quotient Ring of External Direct Sum of Rings",
"Quotient Ring ... |
proofwiki-6594 | Epsilon-Function Differentiability Condition | Let $f: D \to \R$ be a continuous function, where $D \subseteq \R$ is an open set.
Let $x \in \R$.
Then $f$ is differentiable at $x$ {{iff}} there exist $\alpha \in \R$ and $r \in \R_{>0}$ such that for all $h \in \openint {-r} r \setminus \set 0$:
:$\map f {x + h} = \map f x + h \paren {\alpha + \map \epsilon h}$
wher... | === Necessary Condition ===
Assume that $f$ is differentiable in $x$.
By definition of open set, there exists $r \in \R_{>0}$ such that $\openint {x - r} {x + r} \subseteq D$.
Define $\epsilon: \openint {-r} r \setminus \set 0 \to \R$ by:
:$\map \epsilon h = \dfrac {\map f {x + h} - \map f x} h - \map {f'} x$
If $h \in... | Let $f: D \to \R$ be a [[Definition:Continuous Real Function|continuous function]], where $D \subseteq \R$ is an [[Definition:Open Set (Real Analysis)|open set]].
Let $x \in \R$.
Then $f$ is [[Definition:Differentiable Real Function|differentiable]] at $x$ {{iff}} there exist $\alpha \in \R$ and $r \in \R_{>0}$ such... | === Necessary Condition ===
Assume that $f$ is [[Definition:Differentiable Real Function|differentiable]] in $x$.
By [[Definition:Open Set (Real Analysis)|definition of open set]], there exists $r \in \R_{>0}$ such that $\openint {x - r} {x + r} \subseteq D$.
Define $\epsilon: \openint {-r} r \setminus \set 0 \to \R... | Epsilon-Function Differentiability Condition | https://proofwiki.org/wiki/Epsilon-Function_Differentiability_Condition | https://proofwiki.org/wiki/Epsilon-Function_Differentiability_Condition | [
"Differential Calculus",
"Epsilon-Function Differentiability Condition"
] | [
"Definition:Continuous Real Function",
"Definition:Open Set/Real Analysis",
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Interval/Open",
"Definition:Real Function",
"Definition:Continuously Differentiable/Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Open Set/Real Analysis",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Continuously Differentiable/Real Function",
"Composite of Continuous Mappings is Continuous/Corollary",
"Definition:Continuous Real Function",
"Def... |
proofwiki-6595 | Von Neumann Hierarchy is Cumulative | Let $x$ and $y$ be ordinals such that $x < y$.
Then:
:$\map V x \subsetneqq \map V y$ | By Von Neumann Hierarchy Comparison:
:$(1): \quad \map V x \in \map V y$
By $(1)$ and the {{Axiom-link|Foundation}}:
:$\map V x \ne \map V y$
Furthermore, by $(1)$ and Von Neumann Hierarchy is Supertransitive:
{{finish}}
Category:Von Neumann Hierarchy
5n724nfrxdy07evgcu0uyykv75e34di | Let $x$ and $y$ be [[Definition:Ordinal|ordinals]] such that $x < y$.
Then:
:$\map V x \subsetneqq \map V y$ | By [[Von Neumann Hierarchy Comparison]]:
:$(1): \quad \map V x \in \map V y$
By $(1)$ and the {{Axiom-link|Foundation}}:
:$\map V x \ne \map V y$
Furthermore, by $(1)$ and [[Von Neumann Hierarchy is Supertransitive]]:
{{finish}}
[[Category:Von Neumann Hierarchy]]
5n724nfrxdy07evgcu0uyykv75e34di | Von Neumann Hierarchy is Cumulative | https://proofwiki.org/wiki/Von_Neumann_Hierarchy_is_Cumulative | https://proofwiki.org/wiki/Von_Neumann_Hierarchy_is_Cumulative | [
"Von Neumann Hierarchy"
] | [
"Definition:Ordinal"
] | [
"Von Neumann Hierarchy Comparison",
"Von Neumann Hierarchy is Supertransitive",
"Category:Von Neumann Hierarchy"
] |
proofwiki-6596 | Limits of Real and Imaginary Parts | Let $f: D \to \C$ be a complex function, where $D \subseteq \C$.
Let $z_0 \in D$ be a complex number.
Suppose $f$ is continuous at $z_0$.
Then:
:$(1): \quad \ds \lim_{z \mathop \to z_0} \map \Re {\map f z} = \map \Re {\lim_{z \mathop \to z_0} \map f z}$
:$(2): \quad \ds \lim_{z \mathop \to z_0} \map \Im {\map f z} = \m... | By definition of continuity:
:$\forall \epsilon > 0: \exists \delta > 0: \cmod {z - z_0} < \delta \implies \cmod {\map f z - \map f {z_0} } < \epsilon$
Given $\epsilon > 0$, we find $\delta > 0$ so for all $z \in \C$ with $\cmod {z - z_0} < \delta$:
{{begin-eqn}}
{{eqn | l = \epsilon
| o = >
| r = \cmod {\m... | Let $f: D \to \C$ be a [[Definition:Complex Function|complex function]], where $D \subseteq \C$.
Let $z_0 \in D$ be a [[Definition:Complex Number|complex number]].
Suppose $f$ is [[Definition:Continuous Complex Function|continuous]] at $z_0$.
Then:
:$(1): \quad \ds \lim_{z \mathop \to z_0} \map \Re {\map f z} = \... | By [[Definition:Continuous Complex Function#Epsilon-Delta Definition|definition of continuity]]:
:$\forall \epsilon > 0: \exists \delta > 0: \cmod {z - z_0} < \delta \implies \cmod {\map f z - \map f {z_0} } < \epsilon$
Given $\epsilon > 0$, we find $\delta > 0$ so for all $z \in \C$ with $\cmod {z - z_0} < \delta$:... | Limits of Real and Imaginary Parts | https://proofwiki.org/wiki/Limits_of_Real_and_Imaginary_Parts | https://proofwiki.org/wiki/Limits_of_Real_and_Imaginary_Parts | [
"Limits of Complex Functions"
] | [
"Definition:Complex Function",
"Definition:Complex Number",
"Definition:Continuous Complex Function",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part"
] | [
"Definition:Continuous Complex Function",
"Modulus Larger than Real Part",
"Addition of Real and Imaginary Parts",
"Definition:Complex Number/Imaginary Part",
"Category:Limits of Complex Functions"
] |
proofwiki-6597 | Odd Number Theorem/Corollary | A recurrence relation for the square numbers is:
:$S_n = S_{n - 1} + 2 n - 1$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{j \mathop = 1}^n \paren {2 j - 1}
| c = Odd Number Theorem
}}
{{eqn | r = \sum_{j \mathop = 1}^{n - 1} \paren {2 j - 1} + \paren {2 n - 1}
| c = {{Defof|Summation}}
}}
{{eqn | r = S_{n - 1} + \paren {2 n - 1}
| c = Odd Number Theorem
}}
{{end-eqn}}
{{qed... | A [[Definition:Recurrence Relation|recurrence relation]] for the [[Definition:Square Number|square numbers]] is:
:$S_n = S_{n - 1} + 2 n - 1$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{j \mathop = 1}^n \paren {2 j - 1}
| c = [[Odd Number Theorem]]
}}
{{eqn | r = \sum_{j \mathop = 1}^{n - 1} \paren {2 j - 1} + \paren {2 n - 1}
| c = {{Defof|Summation}}
}}
{{eqn | r = S_{n - 1} + \paren {2 n - 1}
| c = [[Odd Number Theorem]]
}}
{{end-eqn... | Odd Number Theorem/Corollary | https://proofwiki.org/wiki/Odd_Number_Theorem/Corollary | https://proofwiki.org/wiki/Odd_Number_Theorem/Corollary | [
"Odd Number Theorem"
] | [
"Definition:Recursive Sequence/Recurrence Relation",
"Definition:Square Number"
] | [
"Odd Number Theorem",
"Odd Number Theorem",
"Category:Odd Number Theorem"
] |
proofwiki-6598 | Effect of Sequence of Elementary Row Operations on Determinant | Let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operations.
Here, $\hat o_i$ denotes an elementary row operation on a square matrix of order $n$ over a commutative ring with unity $\struct {R, +, \circ}$.
Here, $i \in \set {1, \ldots, m}$.
Then there exists $c \in R$ such that for all square ma... | Proof by induction on $m$, the number of elementary row operations in the sequence $\hat o_1, \ldots, \hat o_m$. | Let $\hat o_1, \ldots, \hat o_m$ be a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Row Operation|elementary row operations]].
Here, $\hat o_i$ denotes an [[Definition:Elementary Row Operation|elementary row operation]] on a [[Definition:Square Matrix|square matrix]] of [[Definition:Order ... | Proof by [[Principle of Mathematical Induction|induction]] on $m$, the number of [[Definition:Elementary Row Operation|elementary row operations]] in the [[Definition:Finite Sequence|sequence]] $\hat o_1, \ldots, \hat o_m$. | Effect of Sequence of Elementary Row Operations on Determinant | https://proofwiki.org/wiki/Effect_of_Sequence_of_Elementary_Row_Operations_on_Determinant | https://proofwiki.org/wiki/Effect_of_Sequence_of_Elementary_Row_Operations_on_Determinant | [
"Determinants",
"Elementary Row Operations",
"Proofs by Induction"
] | [
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Commutative Ring",
"Definition:Ring with Unity",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/S... | [
"Principle of Mathematical Induction",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Fin... |
proofwiki-6599 | Every Filter has Adherent Point implies Every Ultrafilter Converges | Let $T = \struct {S, \tau}$ be a topological space.
Let every filter on $S$ have a adherent point in $S$.
Then every ultrafilter on $S$ converges to a point in $S$. | {{Recall|Ultrafilter on Set|ultrafilter}}
{{:Definition:Ultrafilter on Set/Definition 1}}
Let $T = \struct {S, \tau}$ be such that every filter on $S$ has a adherent point in $S$.
Let $\FF$ be an arbitrary ultrafilter on $S$.
We have {{hypothesis}} that $\FF$ has a adherent point $x \in S$.
By Adherent Point of Filter ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let every [[Definition:Filter on Set|filter]] on $S$ have a [[Definition:Adherent Point of Filter|adherent point]] in $S$.
Then every [[Definition:Ultrafilter on Set|ultrafilter]] on $S$ [[Definition:Convergent Filter|converges]] to... | {{Recall|Ultrafilter on Set|ultrafilter}}
{{:Definition:Ultrafilter on Set/Definition 1}}
Let $T = \struct {S, \tau}$ be such that every [[Definition:Filter on Set|filter]] on $S$ has a [[Definition:Adherent Point of Filter|adherent point]] in $S$.
Let $\FF$ be an [[Definition:Arbitrary|arbitrary]] [[Definition:Ultra... | Every Filter has Adherent Point implies Every Ultrafilter Converges | https://proofwiki.org/wiki/Every_Filter_has_Adherent_Point_implies_Every_Ultrafilter_Converges | https://proofwiki.org/wiki/Every_Filter_has_Adherent_Point_implies_Every_Ultrafilter_Converges | [
"Filters on Sets",
"Ultrafilters on Sets",
"Adherent Points",
"Convergent Filters"
] | [
"Definition:Topological Space",
"Definition:Filter on Set",
"Definition:Adherent Point/Filter",
"Definition:Ultrafilter on Set",
"Definition:Convergent Filter",
"Definition:Element"
] | [
"Definition:Filter on Set",
"Definition:Adherent Point/Filter",
"Definition:Arbitrary",
"Definition:Ultrafilter on Set",
"Definition:Adherent Point/Filter",
"Adherent Point of Filter iff Superfilter Converges",
"Definition:Filter",
"Definition:Convergent Filter",
"Definition:Ultrafilter on Set",
"... |
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