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-13300 | Strictly Precede and Step Condition and not Precede implies Joins are equal | Let $\struct {S, \vee, \preceq}$ be a join semilattice.
Let $p, q, u \in S$ be such that:
:$p \prec q$ and $\paren {\forall s \in S: p \prec s \implies q \preceq s}$ and $u \npreceq p$
Then:
:$p \vee u = q \vee u$ | From the definition of join, it is required to prove that:
:$\forall s \in S: p \preceq s \land u \preceq s \implies q \vee u \preceq s$
Let $s \in S$ be such that:
:$p \preceq s$ and $u \preceq s$
We have:
:$p \ne s$
By definition of strictly precede:
:$p \prec s$
By assumption:
:$q \preceq s$
Thus by definition of th... | Let $\struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $p, q, u \in S$ be such that:
:$p \prec q$ and $\paren {\forall s \in S: p \prec s \implies q \preceq s}$ and $u \npreceq p$
Then:
:$p \vee u = q \vee u$ | From the definition of [[Definition:Join (Order Theory)|join]], it is required to prove that:
:$\forall s \in S: p \preceq s \land u \preceq s \implies q \vee u \preceq s$
Let $s \in S$ be such that:
:$p \preceq s$ and $u \preceq s$
We have:
:$p \ne s$
By definition of [[Definition:Strictly Precede|strictly precede... | Strictly Precede and Step Condition and not Precede implies Joins are equal | https://proofwiki.org/wiki/Strictly_Precede_and_Step_Condition_and_not_Precede_implies_Joins_are_equal | https://proofwiki.org/wiki/Strictly_Precede_and_Step_Condition_and_not_Precede_implies_Joins_are_equal | [
"Join and Meet Semilattices"
] | [
"Definition:Join Semilattice"
] | [
"Definition:Join (Order Theory)",
"Definition:Strictly Precede",
"Definition:Join (Order Theory)"
] |
proofwiki-13301 | Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Let $x, y \in S$ such that
:$y \npreceq x$
Then
:$\exists p \in S: p$ is completely irreducible $\mathop \land x \preceq p \land y \npreceq p$ | By definition of algebraic:
:$\forall z \in S: z^\ll$ is directed
and
:$L$ satisfies the axiom of approximation.
By Axiom of Approximation in Up-Complete Semilattice:
:$\exists k \in S: k \ll y \land k \npreceq x$
By Algebraic iff Continuous and For Every Way Below Exists Compact Between:
:$\exists z \in \map K L: k \p... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Let $x, y \in S$ such that
:$y \npreceq x$
Then
:$\exists p \in S: p$ is [[Definition:Completely Irreducible|completely irredu... | By definition of [[Definition:Algebraic Ordered Set|algebraic]]:
:$\forall z \in S: z^\ll$ is [[Definition:Directed Subset|directed]]
and
:$L$ satisfies the [[Axiom:Axiom of Approximation|axiom of approximation]].
By [[Axiom of Approximation in Up-Complete Semilattice]]:
:$\exists k \in S: k \ll y \land k \npreceq x$
... | Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice | https://proofwiki.org/wiki/Not_Preceding_implies_Exists_Completely_Irreducible_Element_in_Algebraic_Lattice | https://proofwiki.org/wiki/Not_Preceding_implies_Exists_Completely_Irreducible_Element_in_Algebraic_Lattice | [
"Continuous Lattices",
"Meet Irreducible Elements"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Completely Irreducible"
] | [
"Definition:Algebraic Ordered Set",
"Definition:Directed Subset",
"Axiom:Axiom of Approximation",
"Axiom of Approximation in Up-Complete Semilattice",
"Algebraic iff Continuous and For Every Way Below Exists Compact Between",
"Definition:Transitive",
"Definition:Upper Closure/Element",
"Definition:Set... |
proofwiki-13302 | Set of All Completely Irreducible Elements is Smallest Order Generating | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Then $\map {\operatorname {Irr} } L$ is order generating and
:$\forall X \subseteq S: X$ is order generating $\implies \map {\operatorname {Irr} } L \subseteq X$
where $\map {\operatorname {Irr} } L$ denotes the set of all completely irr... | By Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice:
:$\forall x, y \in S: y \npreceq x \implies \exists p \in \map {\operatorname {Irr} } L: p \preceq x \land p \npreceq y$
Thus by Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding:
:$\map {\oper... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Then $\map {\operatorname {Irr} } L$ is [[Definition:Order Generating Subset|order generating]] and
:$\forall X \subseteq S: X$... | By [[Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice]]:
:$\forall x, y \in S: y \npreceq x \implies \exists p \in \map {\operatorname {Irr} } L: p \preceq x \land p \npreceq y$
Thus by [[Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding]]:
:$\m... | Set of All Completely Irreducible Elements is Smallest Order Generating | https://proofwiki.org/wiki/Set_of_All_Completely_Irreducible_Elements_is_Smallest_Order_Generating | https://proofwiki.org/wiki/Set_of_All_Completely_Irreducible_Elements_is_Smallest_Order_Generating | [
"Meet Irreducible Elements",
"Order Generating Subsets"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Order Generating Subset",
"Definition:Order Generating Subset",
"Definition:Set",
"Definition:Completely Irreducible"
] | [
"Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice",
"Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding",
"Definition:Order Generating Subset",
"Definition:Order Generating Subset",
"Definition:Completely Irreducible",
"Order Generat... |
proofwiki-13303 | Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $X \subseteq S$, $p \in S$ such that
:$p$ is completely irreducible and $p = \inf X$
Then $p \in X$ | {{AimForCont}}
:$p \notin X$
By Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element:
:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^\succeq$
where $p^\succeq$ denotes the upper closure of $... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $X \subseteq S$, $p \in S$ such that
:$p$ is [[Definition:Completely Irreducible|completely irreducible]] and $p = \inf X$
Then $p \in X$ | {{AimForCont}}
:$p \notin X$
By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]:
:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^\succeq$
where $p^\succeq$ denotes the [[Definition:... | Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset | https://proofwiki.org/wiki/Completely_Irreducible_Element_equals_Infimum_of_Subset_implies_Element_Belongs_to_Subset | https://proofwiki.org/wiki/Completely_Irreducible_Element_equals_Infimum_of_Subset_implies_Element_Belongs_to_Subset | [
"Meet Irreducible Elements"
] | [
"Definition:Complete Lattice",
"Definition:Completely Irreducible"
] | [
"Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element",
"Definition:Upper Closure/Element",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Strictly Precede",
"Definition:Strictly Precede",
"Definition:Lower B... |
proofwiki-13304 | Preimage of Lower Section under Increasing Mapping is Lower | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets.
Let $f: S \to T$ be an increasing mapping.
Let $X \subseteq T$ be a lower section of $T$.
Then:
:$f^{-1} \sqbrk X$ is lower
where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$. | Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that $y \preceq x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \preceq
| r = x
}}
{{eqn | ll= \leadsto
| l = \map f y
| o = \precsim
| r = \map f x
| c = {{Defof|Increasing Mapping}}
}}
{{eqn | l = x
| o = \in
| r = f^{-1} \sqbrk X... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Preordered Set|preordered sets]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $X \subseteq T$ be a [[Definition:Lower Section|lower section]] of $T$.
Then:
:$f^{-1} \sqbrk X$ is [[Definition:Lower Section|lower]... | Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that $y \preceq x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \preceq
| r = x
}}
{{eqn | ll= \leadsto
| l = \map f y
| o = \precsim
| r = \map f x
| c = {{Defof|Increasing Mapping}}
}}
{{eqn | l = x
| o = \in
| r = f^{-1} \sqbrk ... | Preimage of Lower Section under Increasing Mapping is Lower | https://proofwiki.org/wiki/Preimage_of_Lower_Section_under_Increasing_Mapping_is_Lower | https://proofwiki.org/wiki/Preimage_of_Lower_Section_under_Increasing_Mapping_is_Lower | [
"Lower Sections"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Increasing/Mapping",
"Definition:Lower Section",
"Definition:Lower Section",
"Definition:Preimage/Mapping/Subset"
] | [] |
proofwiki-13305 | Preimage of Upper Section under Increasing Mapping is Upper | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets.
Let $f: S \to T$ be an increasing mapping.
Let $X \subseteq T$ be a upper subset of $T$.
Then $f^{-1} \sqbrk X$ is upper
where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$. | Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that
:$x \preceq y$
By definition of increasing mapping:
:$\map f x \precsim \map f y$
By definition of preimage of set:
:$\map f x \in X$
By definition of upper section:
:$\map f y \in X$
Thus by definition of preimage of subset:
:$y \in f^{-1} \sqbrk X$
{{qed}} | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Preordered Set|preordered sets]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $X \subseteq T$ be a [[Definition:Upper Section|upper]] [[Definition:Subset|subset]] of $T$.
Then $f^{-1} \sqbrk X$ is [[Definition:U... | Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that
:$x \preceq y$
By definition of [[Definition:Increasing Mapping|increasing mapping]]:
:$\map f x \precsim \map f y$
By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
:$\map f x \in X$
By definition of [[Definition:Upper Section|upper se... | Preimage of Upper Section under Increasing Mapping is Upper | https://proofwiki.org/wiki/Preimage_of_Upper_Section_under_Increasing_Mapping_is_Upper | https://proofwiki.org/wiki/Preimage_of_Upper_Section_under_Increasing_Mapping_is_Upper | [
"Upper Sections"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Increasing/Mapping",
"Definition:Upper Section",
"Definition:Subset",
"Definition:Upper Section",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Increasing/Mapping",
"Definition:Preimage/Mapping/Subset",
"Definition:Upper Section",
"Definition:Preimage/Mapping/Subset"
] |
proofwiki-13306 | Continuous implies Increasing in Scott Topological Lattices | Let $T_1 = \struct {S_1, \preceq_1, \tau_1}$ and $T_2 = \struct {S_2, \preceq_2, \tau_2}$ be up-complete topological lattices with Scott topologies.
Let $f: S_1 \to S_2$ be a continuous mapping.
Then $f$ is an increasing mapping. | Let $x, y \in S_1$ such that
:$x \preceq_1 y$
{{AimForCont}} that
:$\map f x \npreceq_2 \map f y$
By definition of lower closure of element:
:$\map f x \notin \paren {\map f y}^\preceq$
By definition of relative complement:
:$\map f x \in \relcomp {S_2} {\paren {\map f y}^\preceq}$
By definition of reflexivity:
:$\map ... | Let $T_1 = \struct {S_1, \preceq_1, \tau_1}$ and $T_2 = \struct {S_2, \preceq_2, \tau_2}$ be [[Definition:Up-Complete|up-complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S_1 \to S_2$ be a [[Definition:Continuous (Topology)|continuous]] [[De... | Let $x, y \in S_1$ such that
:$x \preceq_1 y$
{{AimForCont}} that
:$\map f x \npreceq_2 \map f y$
By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
:$\map f x \notin \paren {\map f y}^\preceq$
By definition of [[Definition:Relative Complement|relative complement]]:
:$\map f x \in \re... | Continuous implies Increasing in Scott Topological Lattices | https://proofwiki.org/wiki/Continuous_implies_Increasing_in_Scott_Topological_Lattices | https://proofwiki.org/wiki/Continuous_implies_Increasing_in_Scott_Topological_Lattices | [
"Topological Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Mapping",
"Definition:Increasing/Mapping"
] | [
"Definition:Lower Closure/Element",
"Definition:Relative Complement",
"Definition:Reflexivity",
"Definition:Lower Closure/Element",
"Closure of Singleton is Lower Closure of Element in Scott Topological Lattice",
"Definition:Closure (Topology)",
"Definition:Closed Set/Topology",
"Definition:Closed Set... |
proofwiki-13307 | Directed Suprema Preserving Mapping at Element is Supremum | Let $\struct {S, \vee, \wedge, \preceq}$ and $\struct {T, \vee_2, \wedge_2, \precsim}$ be bounded below continuous lattices.
Let $f: S \to T$ be a mapping such that
:$f$ preserves directed suprema.
Let $x \in S$.
Then:
:$\map f x = \sup \set {\map f w: w \in S \land w \ll x}$ | By definition of continuous:
:$x^\ll$ is directed
and
:$\struct {S, \vee, \wedge, \preceq}$ is up-complete
and
:$\struct {S, \vee, \wedge, \preceq}$ satisfies the {{Axiom-link|Approximation}}.
By definition of mapping preserves directed suprema:
:$f$ preserves the supremum of $x^\ll$.
By definition of up-complete:
:$x^... | Let $\struct {S, \vee, \wedge, \preceq}$ and $\struct {T, \vee_2, \wedge_2, \precsim}$ be [[Definition:Bounded Below Set|bounded below]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Lattice (Order Theory)|lattices]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
:$f$ [[Definition:Mapp... | By definition of [[Definition:Continuous Ordered Set|continuous]]:
:$x^\ll$ is [[Definition:Directed Subset|directed]]
and
:$\struct {S, \vee, \wedge, \preceq}$ is [[Definition:Up-Complete|up-complete]]
and
:$\struct {S, \vee, \wedge, \preceq}$ satisfies the {{Axiom-link|Approximation}}.
By definition of [[Definition:... | Directed Suprema Preserving Mapping at Element is Supremum | https://proofwiki.org/wiki/Directed_Suprema_Preserving_Mapping_at_Element_is_Supremum | https://proofwiki.org/wiki/Directed_Suprema_Preserving_Mapping_at_Element_is_Supremum | [
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Continuous Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Continuous Ordered Set",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set"
] |
proofwiki-13308 | Preceding implies Way Below Closure is Subset of Way Below Closure | Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$ such that
:$x \preceq y$
Then $x^\ll \subseteq y^\ll$
where $x^\ll$ denotes the way below closure of $x$. | Let $z \in x^\ll$.
By definition of way below closure:
:$z \ll x$
By Preceding and Way Below implies Way Below and definition of reflexivity:
:$z \ll y$
Thus by definition of way below closure:
:$z \in y^\ll$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x \preceq y$
Then $x^\ll \subseteq y^\ll$
where $x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$. | Let $z \in x^\ll$.
By definition of [[Definition:Way Below Closure|way below closure]]:
:$z \ll x$
By [[Preceding and Way Below implies Way Below]] and definition of [[Definition:Reflexivity|reflexivity]]:
:$z \ll y$
Thus by definition of [[Definition:Way Below Closure|way below closure]]:
:$z \in y^\ll$
{{qed}} | Preceding implies Way Below Closure is Subset of Way Below Closure | https://proofwiki.org/wiki/Preceding_implies_Way_Below_Closure_is_Subset_of_Way_Below_Closure | https://proofwiki.org/wiki/Preceding_implies_Way_Below_Closure_is_Subset_of_Way_Below_Closure | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Way Below Closure"
] | [
"Definition:Way Below Closure",
"Preceding and Way Below implies Way Below",
"Definition:Reflexivity",
"Definition:Way Below Closure"
] |
proofwiki-13309 | Mapping at Element is Supremum implies Mapping is Increasing | Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
Let $\struct {T, \vee_2, \wedge_2, \precsim}$ be a complete lattice.
Let $f: S \to T$ be a mapping such that:
:$\forall x \in S: \map f x = \sup \set {\map f w: w \in S \land w \ll x}$
Then $f$ is an increasing mapping. | Let $x, y \in S$ such that:
:$x \preceq y$
By Preceding implies Way Below Closure is Subset of Way Below Closure:
:$x^\ll \subseteq y^\ll$
By definitions of image of set and way below closure:
:$f \sqbrk {x^\ll} = \set {\map f w: w \in S \land w \ll x}$
and
:$f \sqbrk {y^\ll} = \set {\map f w: w \in S \land w \ll y}$
w... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $\struct {T, \vee_2, \wedge_2, \precsim}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
:$\forall x \in S: \map f x = \sup \set {\map f w: w \in S \... | Let $x, y \in S$ such that:
:$x \preceq y$
By [[Preceding implies Way Below Closure is Subset of Way Below Closure]]:
:$x^\ll \subseteq y^\ll$
By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Way Below Closure|way below closure]]:
:$f \sqbrk {x^\ll} = \set {\map f w: w \in ... | Mapping at Element is Supremum implies Mapping is Increasing | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_implies_Mapping_is_Increasing | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_implies_Mapping_is_Increasing | [
"Increasing Mappings"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Increasing/Mapping"
] | [
"Preceding implies Way Below Closure is Subset of Way Below Closure",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Way Below Closure",
"Definition:Image (Set Theory)/Mapping/Subset",
"Image of Subset under Mapping is Subset of Image",
"Supremum of Subset"
] |
proofwiki-13310 | Semiperfect Number is not Deficient | Let $n \in \Z_{>0}$ be a semiperfect number.
Then $n$ is not deficient. | Let $n$ be semiperfect.
Then by definition, the sum of the aliquot parts of $n$ is not less than $n$.
The result follows by definition of deficient.
{{qed}}
Category:Semiperfect Numbers
Category:Deficient Numbers
hc85iiuau9aa2yjhlxau95ge4lnn53b | Let $n \in \Z_{>0}$ be a [[Definition:Semiperfect Number|semiperfect number]].
Then $n$ is not [[Definition:Deficient Number|deficient]]. | Let $n$ be [[Definition:Semiperfect Number|semiperfect]].
Then by definition, the [[Definition:Integer Addition|sum]] of the [[Definition:Aliquot Part|aliquot parts]] of $n$ is not less than $n$.
The result follows by definition of [[Definition:Deficient Number|deficient]].
{{qed}}
[[Category:Semiperfect Numbers]]
[... | Semiperfect Number is not Deficient | https://proofwiki.org/wiki/Semiperfect_Number_is_not_Deficient | https://proofwiki.org/wiki/Semiperfect_Number_is_not_Deficient | [
"Semiperfect Numbers",
"Deficient Numbers"
] | [
"Definition:Semiperfect Number",
"Definition:Deficient Number"
] | [
"Definition:Semiperfect Number",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Deficient Number",
"Category:Semiperfect Numbers",
"Category:Deficient Numbers"
] |
proofwiki-13311 | Perfect Number is Primitive Semiperfect | Let $n \in \Z_{>0}$ be a perfect number.
Then $n$ is also a primitive semiperfect number. | Let $n$ be perfect.
From Divisor of Perfect Number is Deficient, all divisors of $n$ are deficient.
But from Semiperfect Number is not Deficient, it follows that the divisors of $n$ cannot be semiperfect.
Hence the result, by definition of primitive semiperfect number.
{{qed}}
Category:Perfect Numbers
Category:Primitiv... | Let $n \in \Z_{>0}$ be a [[Definition:Perfect Number|perfect number]].
Then $n$ is also a [[Definition:Primitive Semiperfect Number|primitive semiperfect number]]. | Let $n$ be [[Definition:Perfect Number|perfect]].
From [[Divisor of Perfect Number is Deficient]], all [[Definition:Divisor of Integer|divisors]] of $n$ are [[Definition:Deficient Number|deficient]].
But from [[Semiperfect Number is not Deficient]], it follows that the [[Definition:Divisor of Integer|divisors]] of $n... | Perfect Number is Primitive Semiperfect | https://proofwiki.org/wiki/Perfect_Number_is_Primitive_Semiperfect | https://proofwiki.org/wiki/Perfect_Number_is_Primitive_Semiperfect | [
"Perfect Numbers",
"Primitive Semiperfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Primitive Semiperfect Number"
] | [
"Definition:Perfect Number",
"Divisor of Perfect Number is Deficient",
"Definition:Divisor (Algebra)/Integer",
"Definition:Deficient Number",
"Semiperfect Number is not Deficient",
"Definition:Divisor (Algebra)/Integer",
"Definition:Semiperfect Number",
"Definition:Primitive Semiperfect Number",
"Ca... |
proofwiki-13312 | Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below | Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a continuous complete lattice.
Let $f: S \to T$ be a mapping such that
:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
Let $x \in S, y \in T$... | By Mapping at Element is Supremum implies Mapping is Increasing:
:$f$ is an increasing mapping. | Let $\left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a [[Definition:Continuous Ordered Set|continuous]] [[Definition:Complete Lattice|complete lattice]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that... | By [[Mapping at Element is Supremum implies Mapping is Increasing]]:
:$f$ is an [[Definition:Increasing Mapping|increasing mapping]]. | Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_implies_Way_Below_iff_There_Exists_Element_that_Way_Below_and_Way_Below | https://proofwiki.org/wiki/Mapping_at_Element_is_Supremum_implies_Way_Below_iff_There_Exists_Element_that_Way_Below_and_Way_Below | [
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Complete Lattice",
"Definition:Mapping"
] | [
"Mapping at Element is Supremum implies Mapping is Increasing",
"Definition:Increasing/Mapping",
"Definition:Increasing/Mapping"
] |
proofwiki-13313 | Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a increasing mapping.
Let $D \subseteq S$ such that
:$D$ admits a supremum in $S$ and $f \sqbrk D$ admits a supremum in $T$.
Then:
:$\map \sup {f \sqbrk D} \precsim \map f {\sup D}$ | By definition of supremum:
:$\sup D$ is upper bound for $D$.
By Increasing Mapping Preserves Upper Bounds:
:$\map f {\sup D}$ is upper bound for $f \sqbrk D$.
Thus by definition of supremum:
:$\map \sup {f \sqbrk D} \precsim \map f {\sup D}$
{{qed}} | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Increasing Mapping|increasing mapping]].
Let $D \subseteq S$ such that
:$D$ admits a [[Definition:Supremum of Set|supremum]] in $S$ and $f \sqbrk D$ admits a [[Definition:Supremum of Set|... | By definition of [[Definition:Supremum of Set|supremum]]:
:$\sup D$ is [[Definition:Upper Bound of Set|upper bound]] for $D$.
By [[Increasing Mapping Preserves Upper Bounds]]:
:$\map f {\sup D}$ is [[Definition:Upper Bound of Set|upper bound]] for $f \sqbrk D$.
Thus by definition of [[Definition:Supremum of Set|supre... | Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum | https://proofwiki.org/wiki/Subset_and_Image_Admit_Suprema_and_Mapping_is_Increasing_implies_Supremum_of_Image_Precedes_Mapping_at_Supremum | https://proofwiki.org/wiki/Subset_and_Image_Admit_Suprema_and_Mapping_is_Increasing_implies_Supremum_of_Image_Precedes_Mapping_at_Supremum | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Increasing Mapping Preserves Upper Bounds",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set"
] |
proofwiki-13314 | Characteristic Subgroup of Normal Subgroup is Normal | Let $G$ be a group.
Let $N\leq G$ be normal.
Let $H\leq N$ be characteristic.
Then $H$ is normal in $G$. | Let $g \in G$.
Because $N$ is normal, conjugation by $g$ is an automorphism of $N$.
Because $H$ is characteristic in $N$, $g H g^{-1} = H$.
Thus $H$ is normal in $G$.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $N\leq G$ be [[Definition:Normal Subgroup|normal]].
Let $H\leq N$ be [[Definition:Characteristic Subgroup|characteristic]].
Then $H$ is [[Definition:Normal Subgroup|normal]] in $G$. | Let $g \in G$.
Because $N$ is [[Definition:Normal Subgroup|normal]], [[Definition:Conjugate of Group Subset|conjugation]] by $g$ is an [[Definition:Group Automorphism|automorphism]] of $N$.
Because $H$ is [[Definition:Characteristic Subgroup|characteristic]] in $N$, $g H g^{-1} = H$.
Thus $H$ is [[Definition:Normal ... | Characteristic Subgroup of Normal Subgroup is Normal | https://proofwiki.org/wiki/Characteristic_Subgroup_of_Normal_Subgroup_is_Normal | https://proofwiki.org/wiki/Characteristic_Subgroup_of_Normal_Subgroup_is_Normal | [
"Characteristic Subgroups",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Characteristic Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Normal Subgroup",
"Definition:Conjugate (Group Theory)/Subset",
"Definition:Group Automorphism",
"Definition:Characteristic Subgroup",
"Definition:Normal Subgroup"
] |
proofwiki-13315 | Center is Characteristic Subgroup | Let $G$ be a group.
Then its center $\map Z G$ is characteristic in $G$. | By Identity Mapping is Group Automorphism, there exists at least one automorphism of $G$.
Let $\phi$ be an automorphism of $G$.
Let $x \in \map Z G, y \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x y
| r = \map \phi x \map \phi {\map {\phi^{-1} } y}
| c = automorphisms are bijections
}}
{{eqn | r = \ma... | Let $G$ be a [[Definition:Group|group]].
Then its [[Definition:Center of Group|center]] $\map Z G$ is [[Definition:Characteristic Subgroup|characteristic]] in $G$. | By [[Identity Mapping is Group Automorphism]], there exists at least one [[Definition:Automorphism (Abstract Algebra)|automorphism]] of $G$.
Let $\phi$ be an [[Definition:Automorphism (Abstract Algebra)|automorphism]] of $G$.
Let $x \in \map Z G, y \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x y
| r = \m... | Center is Characteristic Subgroup | https://proofwiki.org/wiki/Center_is_Characteristic_Subgroup | https://proofwiki.org/wiki/Center_is_Characteristic_Subgroup | [
"Characteristic Subgroups",
"Centers of Groups"
] | [
"Definition:Group",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Characteristic Subgroup"
] | [
"Identity Mapping is Automorphism/Groups",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Bijection",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Bijection",
"Definition:Characteristic ... |
proofwiki-13316 | Semidirect Product of Groups is Group | Let $H$ and $N$ be groups.
Let $\Aut N$ denote the automorphism group of $N$.
Let $\phi : H\to \Aut N$ be a group homomorphism, that is, let $H$ act on $N$.
Let $N \rtimes_\phi H$ be the (outer)semidirect product of $N$ and $H$ with respect to $\phi$, that is:
:$N \rtimes_\phi H = (N \times H, \circ)$ where
:$(n_1, h_1... | === Associativity ===
Let $(n_1,h_1),(n_2,h_2),(n_3,h_3)\in N\times H$. Then
{{begin-eqn}}
{{eqn | l = ((n_1, h_1) \circ (n_2, h_2)) \circ (n_3, h_3)
| r = ((n_1\cdot \phi_{h_1}(n_2), h_1\cdot h_2)) \circ (n_3,h_3)
}}
{{eqn | l =
| r = (n_1\cdot \phi_{h_1}(n_2) \cdot \phi_{h_1h_2}(n_3), h_1\cdot h_2\cdot h... | Let $H$ and $N$ be [[Definition:Group|groups]].
Let $\Aut N$ denote the [[Definition:Automorphism Group of Group|automorphism group]] of $N$.
Let $\phi : H\to \Aut N$ be a [[Definition:Group Homomorphism|group homomorphism]], that is, let $H$ [[Definition:Group Action|act]] on $N$.
Let $N \rtimes_\phi H$ be the [[De... | === Associativity ===
Let $(n_1,h_1),(n_2,h_2),(n_3,h_3)\in N\times H$. Then
{{begin-eqn}}
{{eqn | l = ((n_1, h_1) \circ (n_2, h_2)) \circ (n_3, h_3)
| r = ((n_1\cdot \phi_{h_1}(n_2), h_1\cdot h_2)) \circ (n_3,h_3)
}}
{{eqn | l =
| r = (n_1\cdot \phi_{h_1}(n_2) \cdot \phi_{h_1h_2}(n_3), h_1\cdot h_2\cdot... | Semidirect Product of Groups is Group | https://proofwiki.org/wiki/Semidirect_Product_of_Groups_is_Group | https://proofwiki.org/wiki/Semidirect_Product_of_Groups_is_Group | [
"Semidirect Products"
] | [
"Definition:Group",
"Definition:Automorphism Group/Group",
"Definition:Group Homomorphism",
"Definition:Group Action",
"Definition:Semidirect Product/Outer",
"Definition:Group"
] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Group Automorphism"
] |
proofwiki-13317 | Inverse of Element in Semidirect Product | Let $N$ and $H$ be groups.
Let $H$ act by automorphisms on $N$ via $\phi$.
Let $N \rtimes_\phi H$ be the corresponding (outer) semidirect product.
Let $\tuple {n, h} \in N \rtimes_\phi H$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {n, h}^{-1}
| r = \tuple {\map {\phi_{h^{-1} } } {n^{-1} }, h^{-1} }
| c =
}}
{... | Follows from Semidirect Product of Groups is Group.
The alternatives follow from the fact that $H$ acts by automorphisms.
{{qed}}
{{finish|Expand the proof by demonstrating how it works.}}
Category:Group Theory
Category:Semidirect Products
o8rs9bgvrqwle1hig2ok7q7zmyknuue | Let $N$ and $H$ be [[Definition:Group|groups]].
Let $H$ [[Definition:Group Action by Automorphisms|act by automorphisms]] on $N$ via $\phi$.
Let $N \rtimes_\phi H$ be the corresponding [[Definition:Outer Semidirect Product|(outer) semidirect product]].
Let $\tuple {n, h} \in N \rtimes_\phi H$.
Then:
{{begin-eqn}}... | Follows from [[Semidirect Product of Groups is Group]].
The alternatives follow from the fact that $H$ [[Definition:Group Action by Automorphisms|acts by automorphisms]].
{{qed}}
{{finish|Expand the proof by demonstrating how it works.}}
[[Category:Group Theory]]
[[Category:Semidirect Products]]
o8rs9bgvrqwle1hig2ok... | Inverse of Element in Semidirect Product | https://proofwiki.org/wiki/Inverse_of_Element_in_Semidirect_Product | https://proofwiki.org/wiki/Inverse_of_Element_in_Semidirect_Product | [
"Group Theory",
"Semidirect Products"
] | [
"Definition:Group",
"Definition:Group Action by Automorphisms",
"Definition:Semidirect Product/Outer"
] | [
"Semidirect Product of Groups is Group",
"Definition:Group Action by Automorphisms",
"Category:Group Theory",
"Category:Semidirect Products"
] |
proofwiki-13318 | Semidirect Product with Trivial Action is Direct Product | Let $H$ and $N$ be groups.
Let $\Aut N$ denote the automorphism group of $N$.
Let $\phi: H \to \Aut N$ be defined as:
:$\forall h \in H: \map \phi h = I_N$ for all $h \in H$
where $I_N$ denotes the identity mapping on $N$.
Let $N \rtimes_\phi H$ be the corresponding semidirect product.
Then $N \rtimes_\phi H$ is the di... | Pick arbitrary $\tuple {n_1, h_1}, \tuple {n_2, h_2} \in N \rtimes_\phi H$.
{{begin-eqn}}
{{eqn | l = \tuple {n_1, h_1} \tuple {n_2, h_2}
| r = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 h_2}
| c = {{Defof|Outer Semidirect Product}}
}}
{{eqn | r = \tuple {n_1 \cdot \map {I_N} {n_2}, h_1 h_2}
... | Let $H$ and $N$ be [[Definition:Group|groups]].
Let $\Aut N$ denote the [[Definition:Automorphism Group of Group|automorphism group]] of $N$.
Let $\phi: H \to \Aut N$ be defined as:
:$\forall h \in H: \map \phi h = I_N$ for all $h \in H$
where $I_N$ denotes the [[Definition:Identity Mapping|identity mapping]] on $N$.... | Pick arbitrary $\tuple {n_1, h_1}, \tuple {n_2, h_2} \in N \rtimes_\phi H$.
{{begin-eqn}}
{{eqn | l = \tuple {n_1, h_1} \tuple {n_2, h_2}
| r = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 h_2}
| c = {{Defof|Outer Semidirect Product}}
}}
{{eqn | r = \tuple {n_1 \cdot \map {I_N} {n_2}, h_1 h_2}
... | Semidirect Product with Trivial Action is Direct Product | https://proofwiki.org/wiki/Semidirect_Product_with_Trivial_Action_is_Direct_Product | https://proofwiki.org/wiki/Semidirect_Product_with_Trivial_Action_is_Direct_Product | [
"Semidirect Products"
] | [
"Definition:Group",
"Definition:Automorphism Group/Group",
"Definition:Identity Mapping",
"Definition:Semidirect Product/Outer",
"Definition:External Direct Product"
] | [
"Definition:External Direct Product",
"Category:Semidirect Products"
] |
proofwiki-13319 | Semidirect Product is Abelian iff Components are Abelian and Action is Trivial | Let $N$ and $H$ be groups.
Let $H$ act by automorphisms on $N$ via $\phi$.
Let $N \rtimes_\phi H$ be the corresponding (outer) semidirect product.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$N \rtimes_\phi H$ is abelian}}
{{item|(2):|$N$ and $H$ are abelian and $H$ acts trivially}}
{{end-itemize}} | === $(1)$ implies $(2)$ ===
Let $n \in N$, $h \in H$.
From $\tuple {n, e} \tuple {e, h} = \tuple {e,h} \tuple {n, e}$ we have $n \map {\phi_e} e = e \map {\phi_h} n$.
Thus $H$ acts trivially.
By Semidirect Product with Trivial Action is Direct Product, $N \rtimes_\phi H = N \times H$.
By External Direct Product of Abel... | Let $N$ and $H$ be [[Definition:Group|groups]].
Let $H$ [[Definition:Group Action by Automorphisms|act by automorphisms]] on $N$ via $\phi$.
Let $N \rtimes_\phi H$ be the corresponding [[Definition:Outer Semidirect Product|(outer) semidirect product]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$N \rtimes_\phi H$ is [[... | === $(1)$ implies $(2)$ ===
Let $n \in N$, $h \in H$.
From $\tuple {n, e} \tuple {e, h} = \tuple {e,h} \tuple {n, e}$ we have $n \map {\phi_e} e = e \map {\phi_h} n$.
Thus $H$ [[Definition:Trivial Group Action|acts trivially]].
By [[Semidirect Product with Trivial Action is Direct Product]], $N \rtimes_\phi H = N \... | Semidirect Product is Abelian iff Components are Abelian and Action is Trivial | https://proofwiki.org/wiki/Semidirect_Product_is_Abelian_iff_Components_are_Abelian_and_Action_is_Trivial | https://proofwiki.org/wiki/Semidirect_Product_is_Abelian_iff_Components_are_Abelian_and_Action_is_Trivial | [
"Semidirect Products"
] | [
"Definition:Group",
"Definition:Group Action by Automorphisms",
"Definition:Semidirect Product/Outer",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Definition:Trivial Group Action"
] | [
"Definition:Trivial Group Action",
"Semidirect Product with Trivial Action is Direct Product",
"External Direct Product of Abelian Groups is Abelian Group",
"Definition:Abelian Group",
"Semidirect Product with Trivial Action is Direct Product",
"External Direct Product of Abelian Groups is Abelian Group",... |
proofwiki-13320 | Integers whose Squares end in 444 | The sequence of positive integers whose square ends in $444$ begins:
:$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$
{{OEIS|A039685}} | {{begin-eqn}}
{{eqn | l = 38^2
| r = 1444
}}
{{eqn | l = 462^2
| r = 213 \, 444
}}
{{eqn | l = 538^2
| r = 289 \, 444
}}
{{eqn | l = 962^2
| r = 925 \, 444
}}
{{eqn | l = 1038^2
| r = 1 \, 077 \, 444
}}
{{eqn | l = 1462^2
| r = 2 \, 137 \, 444
}}
{{eqn | l = 1538^2
| r = 2 \, 3... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] whose [[Definition:Square (Algebra)|square]] ends in $444$ begins:
:$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$
{{OEIS|A039685}} | {{begin-eqn}}
{{eqn | l = 38^2
| r = 1444
}}
{{eqn | l = 462^2
| r = 213 \, 444
}}
{{eqn | l = 538^2
| r = 289 \, 444
}}
{{eqn | l = 962^2
| r = 925 \, 444
}}
{{eqn | l = 1038^2
| r = 1 \, 077 \, 444
}}
{{eqn | l = 1462^2
| r = 2 \, 137 \, 444
}}
{{eqn | l = 1538^2
| r = 2 \, 3... | Integers whose Squares end in 444 | https://proofwiki.org/wiki/Integers_whose_Squares_end_in_444 | https://proofwiki.org/wiki/Integers_whose_Squares_end_in_444 | [
"Square Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Square/Function"
] | [
"Squares Ending in Repeated Digits",
"Definition:Contradiction",
"Definition:Contradiction"
] |
proofwiki-13321 | Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a increasing mapping.
Let $D \subseteq S$ such that
:$D$ admits a infimum in $S$ and $f \sqbrk D$ admits a infimum in $T$.
Then $\map f {\inf D} \precsim \map \inf {f \sqbrk D}$ | By definition of infimum:
:$\inf D$ is lower bound for $D$.
By Increasing Mapping Preserves Lower Bounds:
:$\map f {\inf D}$ is a lower bound for $f \sqbrk D$.
Thus by definition of infimum:
:$\map f {\inf D} \precsim \map \inf {f \sqbrk D}$
{{qed}} | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Increasing Mapping|increasing mapping]].
Let $D \subseteq S$ such that
:$D$ admits a [[Definition:Infimum of Set|infimum]] in $S$ and $f \sqbrk D$ admits a [[Definition:Infimum of Set|... | By definition of [[Definition:Infimum of Set|infimum]]:
:$\inf D$ is [[Definition:Lower Bound of Set|lower bound]] for $D$.
By [[Increasing Mapping Preserves Lower Bounds]]:
:$\map f {\inf D}$ is a [[Definition:Lower Bound of Set|lower bound]] for $f \sqbrk D$.
Thus by definition of [[Definition:Infimum of Set|infimu... | Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum | https://proofwiki.org/wiki/Subset_and_Image_Admit_Infima_and_Mapping_is_Increasing_implies_Infimum_of_Image_Succeeds_Mapping_at_Infimum | https://proofwiki.org/wiki/Subset_and_Image_Admit_Infima_and_Mapping_is_Increasing_implies_Infimum_of_Image_Succeeds_Mapping_at_Infimum | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping",
"Definition:Infimum of Set",
"Definition:Infimum of Set"
] | [
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Increasing Mapping Preserves Lower Bounds",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set"
] |
proofwiki-13322 | Equivalence of Definitions of Finite Galois Extension | Let $L / K$ be a finite field extension.
{{TFAE|def = Finite Galois Extension}} | === 1 implies 2 ===
Note that by Finite Field Extension has Finite Galois Group, $G = \Aut {L / K}$ is finite.
Let $\alpha \in L$.
Then its orbit under $G$ is finite.
By:
:Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit
its minimal polynomial over $K$ spli... | Let $L / K$ be a [[Definition:Finite Field Extension|finite field extension]].
{{TFAE|def = Finite Galois Extension}} | === 1 implies 2 ===
Note that by [[Finite Field Extension has Finite Galois Group]], $G = \Aut {L / K}$ is [[Definition:Finite Group|finite]].
Let $\alpha \in L$.
Then its [[Definition:Orbit under Group of Permutations|orbit]] under $G$ is [[Definition:Finite Set|finite]].
By:
:[[Minimal Polynomial of Element with ... | Equivalence of Definitions of Finite Galois Extension | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finite_Galois_Extension | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finite_Galois_Extension | [
"Finite Galois Extensions"
] | [
"Definition:Field Extension/Degree/Finite"
] | [
"Finite Field Extension has Finite Galois Group",
"Definition:Finite Group",
"Definition:Orbit under Group of Permutations",
"Definition:Finite Set",
"Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit",
"Definition:Minimal Polynomial",
"Defin... |
proofwiki-13323 | Finite Field Extension has Finite Galois Group | Let $E / F$ be a finite field extension.
Then its Galois group is finite. | Because $E / F$ is finite, it is finitely generated.
Let $\alpha_1, \ldots, \alpha_n \in E$ with $E = \map F {\alpha_1, \ldots, \alpha_n}$.
By Finite Field Extension is Algebraic, $\alpha_1, \ldots, \alpha_n$ are algebraic over $F$.
Let $f_1, \ldots, f_n$ be their minimal polynomials.
Let $f = f_1\dots f_n$.
By Galois ... | Let $E / F$ be a [[Definition:Finite Field Extension|finite field extension]].
Then its [[Definition:Galois Group of Field Extension|Galois group]] is [[Definition:Finite Group|finite]]. | Because $E / F$ is [[Definition:Finite Field Extension|finite]], it is [[Definition:Finitely Generated Field Extension|finitely generated]].
Let $\alpha_1, \ldots, \alpha_n \in E$ with $E = \map F {\alpha_1, \ldots, \alpha_n}$.
By [[Finite Field Extension is Algebraic]], $\alpha_1, \ldots, \alpha_n$ are [[Definition:... | Finite Field Extension has Finite Galois Group | https://proofwiki.org/wiki/Finite_Field_Extension_has_Finite_Galois_Group | https://proofwiki.org/wiki/Finite_Field_Extension_has_Finite_Galois_Group | [
"Field Extensions",
"Galois Groups of Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Galois Group of Field Extension",
"Definition:Finite Group"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Finitely Generated Field Extension",
"Finite Field Extension is Algebraic",
"Definition:Algebraic Element of Field Extension",
"Definition:Minimal Polynomial",
"Galois Group Acts Faithfully on Generating Set",
"Definition:Faithful Group Action",
... |
proofwiki-13324 | Primitive Element Theorem | Let $E / F$ be a separable field extension of finite degree.
Then $E / F$ is simple: there exists $\alpha\in E$ such that $E = \map F \alpha$. | {{tidy}}
{{MissingLinks}}
If $F$ is a finite field (equivalently $E$ is a finite field), this follows from Finite Extension of $\F_p$ is Generated By a Single Element, since the generator of $E / \F_p$ also generates $E / F$.
Next, assume $F$ is infinite.
Choose an algebraic closure $\overline F$ of $F$.
Let $\sigma_1,... | Let $E / F$ be a [[Definition:Separable Field Extension|separable field extension]] of [[Definition:Finite Field Extension|finite degree]].
Then $E / F$ is [[Definition:Simple Field Extension|simple]]: there exists $\alpha\in E$ such that $E = \map F \alpha$. | {{tidy}}
{{MissingLinks}}
If $F$ is a [[Definition:Finite Field|finite field]] (equivalently $E$ is a finite field), this follows from [[Finite Extension of Fp is Generated By a Single Element|Finite Extension of $\F_p$ is Generated By a Single Element]], since the generator of $E / \F_p$ also generates $E / F$.
Next... | Primitive Element Theorem | https://proofwiki.org/wiki/Primitive_Element_Theorem | https://proofwiki.org/wiki/Primitive_Element_Theorem | [
"Field Extensions",
"Named Theorems"
] | [
"Definition:Separable Extension",
"Definition:Field Extension/Degree/Finite",
"Definition:Simple Field Extension"
] | [
"Definition:Galois Field",
"Finite Extension of Fp is Generated By a Single Element",
"Definition:Infinite Field",
"Definition:Algebraic Closure",
"Definition:Embedding (Galois Theory)",
"Definition:Separable Degree/Definition 2",
"Definition:Separable Extension",
"Vector Space over an Infinite Field ... |
proofwiki-13325 | Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers | The largest positive integer which cannot be expressed as the sum of less than $32$ positive fifth powers is $466$:
:$466 = 18 \times 1^5 + 14 \times 2^5$ | {{ProofWanted|It needs to be shown that there are no larger numbers with this property.}} | The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $32$ [[Definition:Positive Integer|positive]] [[Definition:Fifth Power|fifth powers]] is $466$:
:$466 = 18 \times 1^5 + 14 \times 2^5$ | {{ProofWanted|It needs to be shown that there are no larger numbers with this property.}} | Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Less_than_32_Positive_Fifth_Powers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Less_than_32_Positive_Fifth_Powers | [
"Fifth Powers",
"Hilbert-Waring Theorem",
"466"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Fifth Power"
] | [] |
proofwiki-13326 | Smallest Square which is Sum of 3 Fourth Powers | The smallest positive integer whose square is the sum of $3$ fourth powers is $481$:
:$481^2 = 12^4 + 15^4 + 20^4$ | {{begin-eqn}}
{{eqn | l = 12^4 + 15^4 + 20^4
| r = 20 \, 736 + 50 \, 625 + 160 \, 000
| c =
}}
{{eqn | r = 231 \, 361
| c =
}}
{{eqn | r = 481^2
| c =
}}
{{end-eqn}}
The smallest solution must be fourth powers of coprime integers, otherwise dividing by their GCD would yield a smaller solution... | The smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Square (Algebra)|square]] is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fourth Power|fourth powers]] is $481$:
:$481^2 = 12^4 + 15^4 + 20^4$ | {{begin-eqn}}
{{eqn | l = 12^4 + 15^4 + 20^4
| r = 20 \, 736 + 50 \, 625 + 160 \, 000
| c =
}}
{{eqn | r = 231 \, 361
| c =
}}
{{eqn | r = 481^2
| c =
}}
{{end-eqn}}
The smallest solution must be [[Definition:Fourth Power|fourth powers]] of [[Definition:Coprime Integers|coprime integers]], o... | Smallest Square which is Sum of 3 Fourth Powers | https://proofwiki.org/wiki/Smallest_Square_which_is_Sum_of_3_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Square_which_is_Sum_of_3_Fourth_Powers | [
"Square Numbers",
"Fourth Powers"
] | [
"Definition:Positive/Integer",
"Definition:Square/Function",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [
"Definition:Fourth Power",
"Definition:Coprime/Integers",
"Definition:Greatest Common Divisor/Integers",
"Fermat's Right Triangle Theorem",
"Definition:Fourth Power",
"Square Modulo 4",
"Definition:Fourth Power",
"Definition:Square/Function",
"Definition:Odd Integer",
"Definition:Odd Integer",
"... |
proofwiki-13327 | Solutions to p^2 Divides 10^p - 10 | The known prime numbers $p$ which satisfy the equation:
:$p^2 \divides \paren {10^p - 10}$
where $\divides$ denotes divisibility, are:
:$3, 487, 56 \, 598 \, 313$
{{OEIS|A045616}} | {{ProofWanted|Some sort of computer program can be implemented, I suppose}} | The known [[Definition:Prime Number|prime numbers]] $p$ which satisfy the equation:
:$p^2 \divides \paren {10^p - 10}$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]], are:
:$3, 487, 56 \, 598 \, 313$
{{OEIS|A045616}} | {{ProofWanted|Some sort of computer program can be implemented, I suppose}} | Solutions to p^2 Divides 10^p - 10 | https://proofwiki.org/wiki/Solutions_to_p^2_Divides_10^p_-_10 | https://proofwiki.org/wiki/Solutions_to_p^2_Divides_10^p_-_10 | [
"Prime Numbers",
"10"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-13328 | Kaprekar's Process on 3 Digit Number ends in 495 | Let $n$ be a $3$-digit integer whose digits are not all the same.
Kaprekar's process, when applied to $n$, results in $495$ after no more than $6$ iterations. | Let $n = \sqbrk {abc}_{10}$ denote a $3$-digit integer whose digits are $a, b, c$.
If $a = b = c$ then Kaprekar's process trivially results in $0$ after the first iteration.
{{WLOG}}, let $a \ge b \ge c$ but such that $a \ne c$.
By the Basis Representation Theorem:
:$n = 10^2 a + 10 b + c$
Let $n' = 10^2 a' + 10 b' + c... | Let $n$ be a $3$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integer]] whose [[Definition:Digit|digits]] are not all the same.
[[Definition:Kaprekar's Process|Kaprekar's process]], when applied to $n$, results in $495$ after no more than $6$ iterations. | Let $n = \sqbrk {abc}_{10}$ denote a $3$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integer]] whose [[Definition:Digit|digits]] are $a, b, c$.
If $a = b = c$ then [[Definition:Kaprekar's Process|Kaprekar's process]] trivially results in $0$ after the first iteration.
{{WLOG}}, let $a \ge b \ge c$ but s... | Kaprekar's Process on 3 Digit Number ends in 495 | https://proofwiki.org/wiki/Kaprekar's_Process_on_3_Digit_Number_ends_in_495 | https://proofwiki.org/wiki/Kaprekar's_Process_on_3_Digit_Number_ends_in_495 | [
"Kaprekar's Process",
"495"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Kaprekar's Process"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Kaprekar's Process",
"Basis Representation Theorem",
"Definition:Kaprekar's Process",
"Definition:Kaprekar's Process",
"Definition:Zero Digit",
"Definition:Kaprekar's Process"
] |
proofwiki-13329 | Nilpotent Elements of Commutative Ring form Ideal | Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$.
The subset of nilpotent elements of $R$ form an ideal of $R$. | Let $N$ be the subset of nilpotent elements.
Because $0_R$ is nilpotent, $0_R \in N$ and so $N$ is non-empty.
Let $x \in N$ and $a \in R$.
We have:
{{begin-eqn}}
{{eqn | q = \exists n \in \Z_{>0}
| l = x^n
| r = 0_R
| c = {{Defof|Nilpotent Ring Element}}
}}
{{eqn | ll= \leadsto
| l = a^n \circ x... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
The [[Definition:Subset|subset]] of [[Definition:Nilpotent Ring Element|nilpotent elements]] of $R$ form an [[Definition:Ideal of Ring|i... | Let $N$ be the [[Definition:Subset|subset]] of [[Definition:Nilpotent Ring Element|nilpotent elements]].
Because $0_R$ is [[Definition:Nilpotent Ring Element|nilpotent]], $0_R \in N$ and so $N$ is [[Definition:Non-Empty Set|non-empty]].
Let $x \in N$ and $a \in R$.
We have:
{{begin-eqn}}
{{eqn | q = \exists n \in ... | Nilpotent Elements of Commutative Ring form Ideal | https://proofwiki.org/wiki/Nilpotent_Elements_of_Commutative_Ring_form_Ideal | https://proofwiki.org/wiki/Nilpotent_Elements_of_Commutative_Ring_form_Ideal | [
"Nilpotent Ring Elements",
"Commutative Rings",
"Ideal Theory"
] | [
"Definition:Commutative Ring",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Subset",
"Definition:Nilpotent Ring Element",
"Definition:Ideal of Ring"
] | [
"Definition:Subset",
"Definition:Nilpotent Ring Element",
"Definition:Nilpotent Ring Element",
"Definition:Non-Empty Set",
"Power of Product of Commutative Elements in Semigroup",
"Binomial Theorem",
"Test for Ideal",
"Definition:Ideal of Ring"
] |
proofwiki-13330 | Equivalence of Definitions of Nilradical of Ring | {{TFAE|def = Nilradical of Ring}}
Let $A$ be a commutative ring. | By Nilpotent Element is Contained in Prime Ideals, $\Nil A$ is contained in the intersection of all prime ideals.
It remains to prove the other inclusion.
Let $f \in A$ be not nilpotent.
Let $S$ be the set of ideals of $A$ that are disjoint from $\set {f^n: n \in \N}$.
By Zorn's Lemma, $S$ has a maximal element $P$.
In... | {{TFAE|def = Nilradical of Ring}}
Let $A$ be a [[Definition:Commutative Ring|commutative ring]]. | By [[Nilpotent Element is Contained in Prime Ideals]], $\Nil A$ is contained in the [[Definition:Set Intersection|intersection]] of all [[Definition:Prime Ideal of Ring|prime ideals]].
It remains to prove the other inclusion.
Let $f \in A$ be not [[Definition:Nilpotent Ring Element|nilpotent]].
Let $S$ be the [[Def... | Equivalence of Definitions of Nilradical of Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Nilradical_of_Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Nilradical_of_Ring | [
"Nilradicals of Rings"
] | [
"Definition:Commutative Ring"
] | [
"Nilpotent Element is Contained in Prime Ideals",
"Definition:Set Intersection",
"Definition:Prime Ideal of Ring",
"Definition:Nilpotent Ring Element",
"Definition:Set",
"Definition:Ideal of Ring",
"Definition:Disjoint Sets",
"Zorn's Lemma",
"Definition:Maximal/Element",
"Definition:Prime Ideal of... |
proofwiki-13331 | Algebraic Closure of Field is Unique | Let $F$ be a field.
Let $K$ and $L$ be algebraic closures of $F$.
Then $K$ and $L$ are $F$-isomorphic. | {{ProofWanted}}
{{AoC}}
Category:Field Extensions
scjc721k2mx9ewxnymtjblmujpjd2u2 | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $K$ and $L$ be [[Definition:Algebraic Closure|algebraic closures]] of $F$.
Then $K$ and $L$ are [[Definition:F-Isomorphism|$F$-isomorphic]]. | {{ProofWanted}}
{{AoC}}
[[Category:Field Extensions]]
scjc721k2mx9ewxnymtjblmujpjd2u2 | Algebraic Closure of Field is Unique | https://proofwiki.org/wiki/Algebraic_Closure_of_Field_is_Unique | https://proofwiki.org/wiki/Algebraic_Closure_of_Field_is_Unique | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Closure",
"Definition:Isomorphism (Abstract Algebra)/F-Isomorphism"
] | [
"Category:Field Extensions"
] |
proofwiki-13332 | Multiplicative Group of Galois Field is Cyclic | Let $\GF$ be a Galois field of order $q$.
Then its multiplicative group is cyclic of order $q-1$:
:$\GF^\times \cong C_{q - 1}$ | Follows immediately from Finite Multiplicative Subgroup of Field is Cyclic.
{{qed}}
Category:Galois Fields
jjwa16ge80xdni2x02rvbql2smj1753 | Let $\GF$ be a [[Definition:Galois Field|Galois field]] of [[Definition:Order of Structure|order]] $q$.
Then its [[Definition:Multiplicative Group|multiplicative group]] is [[Definition:Cyclic Group|cyclic]] of [[Definition:Order of Structure|order]] $q-1$:
:$\GF^\times \cong C_{q - 1}$ | Follows immediately from [[Finite Multiplicative Subgroup of Field is Cyclic]].
{{qed}}
[[Category:Galois Fields]]
jjwa16ge80xdni2x02rvbql2smj1753 | Multiplicative Group of Galois Field is Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Galois_Field_is_Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Galois_Field_is_Cyclic | [
"Galois Fields"
] | [
"Definition:Galois Field",
"Definition:Order of Structure",
"Definition:Multiplicative Group",
"Definition:Cyclic Group",
"Definition:Order of Structure"
] | [
"Finite Multiplicative Subgroup of Field is Cyclic",
"Category:Galois Fields"
] |
proofwiki-13333 | Automorphism Group of Complex Numbers over Real Numbers | The field extension $\C / \R$ of complex numbers $\C$ over real numbers $\R$ has automorphism group $\operatorname{Aut}$:
:$\operatorname{Aut} \paren {\C / \R} = \set {\operatorname{id}, \sigma}$
where:
:$\operatorname{id}$ denotes the identity mapping
:$\sigma$ denotes complex conjugation | {{ProofWanted}}
Category:Field Extensions
Category:Automorphism Groups
io2w6he36rnjx693tdpc56j0fqej2i5 | The [[Definition:Field Extension|field extension]] $\C / \R$ of [[Definition:Field of Complex Numbers|complex numbers]] $\C$ over [[Definition:Field of Real Numbers|real numbers]] $\R$ has [[Definition:Automorphism Group of Field Extension|automorphism group]] $\operatorname{Aut}$:
:$\operatorname{Aut} \paren {\C / \R}... | {{ProofWanted}}
[[Category:Field Extensions]]
[[Category:Automorphism Groups]]
io2w6he36rnjx693tdpc56j0fqej2i5 | Automorphism Group of Complex Numbers over Real Numbers | https://proofwiki.org/wiki/Automorphism_Group_of_Complex_Numbers_over_Real_Numbers | https://proofwiki.org/wiki/Automorphism_Group_of_Complex_Numbers_over_Real_Numbers | [
"Field Extensions",
"Automorphism Groups"
] | [
"Definition:Field Extension",
"Definition:Field of Complex Numbers",
"Definition:Field of Real Numbers",
"Definition:Galois Group of Field Extension",
"Definition:Identity Mapping",
"Definition:Complex Conjugate/Complex Conjugation"
] | [
"Category:Field Extensions",
"Category:Automorphism Groups"
] |
proofwiki-13334 | Image under Inclusion Mapping | Let $X$ be a set.
Let $S \subseteq X$, $Z \subseteq S$.
Then $i_S \sqbrk Z = Z$
where
:$i_S$ denotes the inclusion mapping of $S$
:$i_S \sqbrk Z$ denotes the image of $Z$ under $i_S$. | {{begin-eqn}}
{{eqn | l = i_S \sqbrk Z
| r = \set {\map {i_S} z: z \in Z}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {z: z \in Z}
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn | r = Z
| c = {{Defof|Set Equality}}
}}
{{end-eqn}}
{{qed}} | Let $X$ be a [[Definition:Set|set]].
Let $S \subseteq X$, $Z \subseteq S$.
Then $i_S \sqbrk Z = Z$
where
:$i_S$ denotes the [[Definition:Inclusion Mapping|inclusion mapping]] of $S$
:$i_S \sqbrk Z$ denotes the [[Definition:Image of Subset under Mapping|image]] of $Z$ under $i_S$. | {{begin-eqn}}
{{eqn | l = i_S \sqbrk Z
| r = \set {\map {i_S} z: z \in Z}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {z: z \in Z}
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn | r = Z
| c = {{Defof|Set Equality}}
}}
{{end-eqn}}
{{qed}} | Image under Inclusion Mapping | https://proofwiki.org/wiki/Image_under_Inclusion_Mapping | https://proofwiki.org/wiki/Image_under_Inclusion_Mapping | [
"Inclusion Mappings"
] | [
"Definition:Set",
"Definition:Inclusion Mapping",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [] |
proofwiki-13335 | Limit Inferior of Inclusion Net is Supremum of Directed Subset | Let $L = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Let $D \subseteq S$ be a directed subset of $S$.
Let $\struct {D, \preceq'}$ be a directed ordered subset of $L$.
Let $i_D: D \to S$, the inclusion mapping, be a net in $S$.
Then $\liminf i_D = \sup D$ | {{Refactor|level = basic|Extract lemma}}
We will prove that:
:(lemma): $\forall j \in D: \map {\inf_L} {\map {\preceq'} j} = j$
Let $j \in D$.
By definitions of image of element and upper closure of element:
:$\map {\preceq'} j = j^{\succeq'}$
By Upper Closure in Ordered Subset is Intersection of Subset and Upper Closu... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Let $D \subseteq S$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Let $\struct {D, \preceq'}$ be a [[Definition:Directed Set|directed]] [[Definition:Ordered Subset|ord... | {{Refactor|level = basic|Extract lemma}}
We will prove that:
:(lemma): $\forall j \in D: \map {\inf_L} {\map {\preceq'} j} = j$
Let $j \in D$.
By definitions of [[Definition:Image of Element under Relation|image of element]] and [[Definition:Upper Closure of Element|upper closure of element]]:
:$\map {\preceq'} j = ... | Limit Inferior of Inclusion Net is Supremum of Directed Subset | https://proofwiki.org/wiki/Limit_Inferior_of_Inclusion_Net_is_Supremum_of_Directed_Subset | https://proofwiki.org/wiki/Limit_Inferior_of_Inclusion_Net_is_Supremum_of_Directed_Subset | [
"Inclusion Mappings",
"Limits Inferior of Nets"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Directed Subset",
"Definition:Directed Preordering",
"Definition:Ordered Subset",
"Definition:Inclusion Mapping",
"Definition:Net (Set Theory)"
] | [
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Upper Closure/Element",
"Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure",
"Intersection is Subset",
"Infimum of Subset",
"Infimum of Upper Closure of Element",
"Definition:Reflexivity",
"Definition:Image (Set Th... |
proofwiki-13336 | Correspondence Between Group Actions and Permutation Representations | Let $G$ be a group.
Let $X$ be a set.
There is a one-to-one correspondence between group actions of $G$ on $X$ and permutation representations of $G$ in $X$, as follows:
Let $\phi : G \times X \to X$ be a group action.
Let $\rho : G \to \struct {\map \Gamma X, \circ}$ be a permutation representation.
The following are ... | For $g\in G$, define the mapping $\phi_g : X \to X$ as:
:$\map {\phi_g} x = \map \phi {g, x}$
Then $\rho$ is the permutation representation associated to $\phi$ {{iff}}:
:$\forall g \in G : \map \rho g = \phi_g$
By Equality of Mappings, this is equivalent to:
:$\forall g \in G : \forall x \in X : \map {\map \rho g} x ... | Let $G$ be a [[Definition:Group|group]].
Let $X$ be a [[Definition:Set|set]].
There is a one-to-one correspondence between [[Definition:Group Action|group actions]] of $G$ on $X$ and [[Definition:Permutation Representation|permutation representations]] of $G$ in $X$, as follows:
Let $\phi : G \times X \to X$ be a [... | For $g\in G$, define the mapping $\phi_g : X \to X$ as:
:$\map {\phi_g} x = \map \phi {g, x}$
Then $\rho$ is the [[Definition:Permutation Representation Associated to Group Action|permutation representation associated to]] $\phi$ {{iff}}:
:$\forall g \in G : \map \rho g = \phi_g$
By [[Equality of Mappings]], this i... | Correspondence Between Group Actions and Permutation Representations | https://proofwiki.org/wiki/Correspondence_Between_Group_Actions_and_Permutation_Representations | https://proofwiki.org/wiki/Correspondence_Between_Group_Actions_and_Permutation_Representations | [
"Group Actions",
"Permutation Representations"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action",
"Definition:Group Representation/Permutation",
"Definition:Group Action",
"Definition:Group Representation/Permutation",
"Definition:Logical Equivalence",
"Definition:Permutation Representation/Group Action",
"Definition:Group Action/P... | [
"Definition:Permutation Representation/Group Action",
"Equality of Mappings",
"Definition:Logical Equivalence",
"Definition:Group Action/Permutation Representation",
"Category:Group Actions",
"Category:Permutation Representations"
] |
proofwiki-13337 | Equivalence of Definitions of Field of Quotients | Let $D$ be an integral domain.
Let $F$ be a field.
{{TFAE|def = Field of Quotients}} | === 1 implies 2 ===
Let $K$ be a field such that:
:$\iota \sqbrk D \subseteq K \subseteq F$
We show that $F \subseteq K$.
Let $f \in F$.
By assumption, there exist $x, y \in D$ with $y \ne 0$ such that $f = \dfrac {\map \iota x} {\map \iota y}$.
Because $K$ is a field containing $\iota \sqbrk D$, $K$ also contains $f =... | Let $D$ be an [[Definition:Integral Domain|integral domain]].
Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
{{TFAE|def = Field of Quotients}} | === 1 implies 2 ===
Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] such that:
:$\iota \sqbrk D \subseteq K \subseteq F$
We show that $F \subseteq K$.
Let $f \in F$.
By assumption, there exist $x, y \in D$ with $y \ne 0$ such that $f = \dfrac {\map \iota x} {\map \iota y}$.
Because $K$ is a [[Definition... | Equivalence of Definitions of Field of Quotients | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Field_of_Quotients | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Field_of_Quotients | [
"Fields of Quotients"
] | [
"Definition:Integral Domain",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract A... |
proofwiki-13338 | Trivial Field Extension is Galois | Let $F$ be a field.
The trivial field extension $F / F$ is Galois. | We shall show {{Defof|Galois Extension/Finite|Galois Extension|index=1}}.
Observe:
{{begin-eqn}}
{{eqn | l = \Gal {F / F}
| r = \set {\sigma \in \Aut F: \forall k \in F: \map \sigma k = k}
| c = {{Defof|Galois Group of Field Extension}}
}}
{{eqn | r = \set {I_F}
| n = 1
}}
{{end-eqn}}
where $I_F$ deno... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
The [[Definition:Trivial Field Extension|trivial field extension]] $F / F$ is [[Definition:Galois Extension|Galois]]. | We shall show {{Defof|Galois Extension/Finite|Galois Extension|index=1}}.
Observe:
{{begin-eqn}}
{{eqn | l = \Gal {F / F}
| r = \set {\sigma \in \Aut F: \forall k \in F: \map \sigma k = k}
| c = {{Defof|Galois Group of Field Extension}}
}}
{{eqn | r = \set {I_F}
| n = 1
}}
{{end-eqn}}
where $I_F$ den... | Trivial Field Extension is Galois | https://proofwiki.org/wiki/Trivial_Field_Extension_is_Galois | https://proofwiki.org/wiki/Trivial_Field_Extension_is_Galois | [
"Galois Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Trivial Field Extension",
"Definition:Galois Extension"
] | [
"Definition:Identity Mapping",
"Category:Galois Theory"
] |
proofwiki-13339 | Field is Galois over Fixed Field of Automorphism Group | Let $E/F$ be a finite field extension.
Let $K = \operatorname{Fix}_E(\operatorname{Aut}(E/F))$ be the fixed field of the automorphism group of $E/F$.
Then $E/K$ is Galois. | Follows from Closed Fields in Galois Connection for Field Extension (and does not use Fundamental Theorem of Galois Theory).
{{proof wanted}}
Category:Galois Theory
5f8c1ibg4vz9kywca8kf6h0d0lu6e4m | Let $E/F$ be a [[Definition:Finite Field Extension|finite field extension]].
Let $K = \operatorname{Fix}_E(\operatorname{Aut}(E/F))$ be the [[Definition:Fixed Field|fixed field]] of the [[Definition:Automorphism Group of Field Extension|automorphism group]] of $E/F$.
Then $E/K$ is [[Definition:Galois Extension|Galoi... | Follows from [[Closed Fields in Galois Connection for Field Extension]] (and does not use [[Fundamental Theorem of Galois Theory]]).
{{proof wanted}}
[[Category:Galois Theory]]
5f8c1ibg4vz9kywca8kf6h0d0lu6e4m | Field is Galois over Fixed Field of Automorphism Group | https://proofwiki.org/wiki/Field_is_Galois_over_Fixed_Field_of_Automorphism_Group | https://proofwiki.org/wiki/Field_is_Galois_over_Fixed_Field_of_Automorphism_Group | [
"Galois Theory"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Fixed Field",
"Definition:Galois Group of Field Extension",
"Definition:Galois Extension"
] | [
"Closed Fields in Galois Connection for Field Extension",
"Fundamental Theorem of Galois Theory",
"Category:Galois Theory"
] |
proofwiki-13340 | Automorphism Group Acts Faithfully on Generating Set | Let $E/F$ be a field extension.
Let $\operatorname{Aut}(E/F)$ be its automorphism group.
Let $S\subset E$ be a generating set of the extension.
Let $S$ be stable under the group action of $\operatorname{Aut}(E/F)$.
Then the induced group action on $S$ is faithful. | Let $\sigma \in \operatorname{Aut}(E/F)$ stabilize $S$.
Then $S$ is contained in the fixed field of $\sigma$.
By definition of generating set, $\sigma$ fixes $E$.
Thus $\operatorname{Aut}(E/F)$ acts faithfully.
{{qed}}
Category:Field Extensions
eqb5qxdw8htgw5zvgpeh6xkvv6l717w | Let $E/F$ be a [[Definition:Field Extension|field extension]].
Let $\operatorname{Aut}(E/F)$ be its [[Definition:Automorphism Group of Field Extension|automorphism group]].
Let $S\subset E$ be a [[Definition:Generator of Field Extension|generating set]] of the extension.
Let $S$ be [[Definition:Stable Under Group Ac... | Let $\sigma \in \operatorname{Aut}(E/F)$ stabilize $S$.
Then $S$ is contained in the [[Definition:Fixed Field|fixed field]] of $\sigma$.
By definition of [[Definition:Generator of Field Extension|generating set]], $\sigma$ fixes $E$.
Thus $\operatorname{Aut}(E/F)$ [[Definition:Faithful Group Action|acts faithfully]]... | Automorphism Group Acts Faithfully on Generating Set | https://proofwiki.org/wiki/Automorphism_Group_Acts_Faithfully_on_Generating_Set | https://proofwiki.org/wiki/Automorphism_Group_Acts_Faithfully_on_Generating_Set | [
"Field Extensions"
] | [
"Definition:Field Extension",
"Definition:Galois Group of Field Extension",
"Definition:Generated Field Extension",
"Definition:Stable Under Group Action",
"Definition:Group Action",
"Definition:Group Action",
"Definition:Faithful Group Action"
] | [
"Definition:Fixed Field",
"Definition:Generated Field Extension",
"Definition:Faithful Group Action",
"Category:Field Extensions"
] |
proofwiki-13341 | Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure | Let $L = \left({S, \preceq}\right)$ be an ordered set.
Let $\left({T, \precsim}\right)$ be an ordered subset of $L$.
Let $t \in T$.
Then $t^\succsim = T \cap t^\succeq$ | By definition of ordered subset:
:$T \subseteq S$
We will prove that
:$t^\succsim \subseteq T \cap t^\succeq$
Let $x \in t^\succsim$
By definition of upper closure of element:
:$x \in T$ and $t \precsim x$
By definition of ordered subset:
:$t \preceq x$
By definition of upper closure of element:
:$x \in t^\succeq$
Thus... | Let $L = \left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $\left({T, \precsim}\right)$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
Let $t \in T$.
Then $t^\succsim = T \cap t^\succeq$ | By definition of [[Definition:Ordered Subset|ordered subset]]:
:$T \subseteq S$
We will prove that
:$t^\succsim \subseteq T \cap t^\succeq$
Let $x \in t^\succsim$
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$x \in T$ and $t \precsim x$
By definition of [[Definition:Ordered Su... | Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure | https://proofwiki.org/wiki/Upper_Closure_in_Ordered_Subset_is_Intersection_of_Subset_and_Upper_Closure | https://proofwiki.org/wiki/Upper_Closure_in_Ordered_Subset_is_Intersection_of_Subset_and_Upper_Closure | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Ordered Subset"
] | [
"Definition:Ordered Subset",
"Definition:Upper Closure/Element",
"Definition:Ordered Subset",
"Definition:Upper Closure/Element",
"Definition:Set Intersection",
"Definition:Set Intersection",
"Definition:Upper Closure/Element",
"Definition:Ordered Subset",
"Definition:Upper Closure/Element",
"Defi... |
proofwiki-13342 | Products of 2-Digit Pairs which Reversed reveal Same Product | The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
:$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$
<!-- fascists won't include the damn th... | Let $n \in \Z_{>0}$ such that:
:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
where $\sqbrk {a b}$ denotes the two-digit positive integer:
: $10 a + b$ for $0 \le a, b \le 9$
from the Basis Representation Theorem.
We have:
{{begin-eqn}}
{{eqn | l = \paren {10 a + b} \paren {10 c + d}
|... | The following [[Definition:Positive Integer|positive integers]] can be expressed as the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Digit|two-digit]] numbers in $2$ ways such that the [[Definition:Divisor of Integer|factors]] in one of those pairs is the [[Definition:Reversal|reversal]] of each of... | Let $n \in \Z_{>0}$ such that:
:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
where $\sqbrk {a b}$ denotes the [[Definition:Digit|two-digit]] [[Definition:Positive Integer|positive integer]]:
: $10 a + b$ for $0 \le a, b \le 9$
from the [[Basis Representation Theorem]].
We have:
{{begin-eq... | Products of 2-Digit Pairs which Reversed reveal Same Product | https://proofwiki.org/wiki/Products_of_2-Digit_Pairs_which_Reversed_reveal_Same_Product | https://proofwiki.org/wiki/Products_of_2-Digit_Pairs_which_Reversed_reveal_Same_Product | [
"Recreational Mathematics",
"Reversals"
] | [
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Reversal",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Basis Representation Theorem",
"Definition:Set",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Divisor Count Function",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Integer",
"Definition:... |
proofwiki-13343 | Approximation to Power of 7 by Power of 10 | :$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$
This is the closest known approximation of a power of $7$ by a power of $10$. | An intuition is given as follows:
Suppose for some $m, n \in \N$:
:$7^m = c \cdot 10^n$, where $c$ is very close to $1$.
Taking common logarithm:
:$m \log 7 = \log c + n$
Which leads to:
:$\log 7 = \dfrac n m + \dfrac {\log c} m$
where $\dfrac {\log c} m$ is very close to $0$.
To make a good approximation is to minimiz... | :$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$
This is the closest known approximation of a [[Definition:Integer Power|power of $7$]] by a [[Definition:Integer Power|power of $10$]]. | An intuition is given as follows:
Suppose for some $m, n \in \N$:
:$7^m = c \cdot 10^n$, where $c$ is very close to $1$.
Taking [[Definition:Common Logarithm|common logarithm]]:
:$m \log 7 = \log c + n$
Which leads to:
:$\log 7 = \dfrac n m + \dfrac {\log c} m$
where $\dfrac {\log c} m$ is very close to $0$.
To m... | Approximation to Power of 7 by Power of 10 | https://proofwiki.org/wiki/Approximation_to_Power_of_7_by_Power_of_10 | https://proofwiki.org/wiki/Approximation_to_Power_of_7_by_Power_of_10 | [
"Powers of 7",
"Powers of 10"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:General Logarithm/Common",
"Definition:Continued Fraction",
"Definition:Continued Fraction",
"Convergents are Best Approximations",
"Definition:Convergent of Continued Fraction",
"Accuracy of Convergents of Continued Fraction",
"Definition:Numerators and Denominators of Continued Fraction"
] |
proofwiki-13344 | Limit Inferior of Repetition Net | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $N = \struct {\N, \le}$ be a directed ordered set.
Let $a, b \in S$.
Let $f = \sequence {c_i}_{i \mathop \in \N} = \tuple {a, b, a, b, \dots}: \N \to S$ be a net.
Then $\liminf \sequence {c_i}_{i \mathop \in \N} = a \wedge b$ | {{refactor|Make this a lemma page|level = basic}}
We will prove that
:(lemma): $\forall j \in \N: f \sqbrk {\le \paren j} = \set {a, b}$
Let $j \in \N$.
Let $x \in S$.
Assume:
:$x \in f \sqbrk {\le \paren j}$
By definition of image of set:
:$\exists i \in \le \paren j: x = \map f i$
By definition of $f$:
:$x = a$ or $x... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $N = \struct {\N, \le}$ be a [[Definition:Directed Set|directed]] [[Definition:Ordered Set|ordered set]].
Let $a, b \in S$.
Let $f = \sequence {c_i}_{i \mathop \in \N} = \tuple {a, b, a, b, \dots}: \N \to S$ be a [[Defi... | {{refactor|Make this a lemma page|level = basic}}
We will prove that
:(lemma): $\forall j \in \N: f \sqbrk {\le \paren j} = \set {a, b}$
Let $j \in \N$.
Let $x \in S$.
Assume:
:$x \in f \sqbrk {\le \paren j}$
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists i \in \le \paren j: ... | Limit Inferior of Repetition Net | https://proofwiki.org/wiki/Limit_Inferior_of_Repetition_Net | https://proofwiki.org/wiki/Limit_Inferior_of_Repetition_Net | [
"Mapping Theory",
"Limits Inferior of Nets"
] | [
"Definition:Meet Semilattice",
"Definition:Directed Preordering",
"Definition:Ordered Set",
"Definition:Net (Set Theory)"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Unordered Tuple",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Unordered Tuple",
"Definition:Set Equality",
"Supremum of Singleton"
] |
proofwiki-13345 | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $f: S \to T$ be a mapping such that:
:for all directed set $\struct {D, \precsim}$ and nets $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
Then $f$ is an increasing mapping. | Let $a, b \in S$ such that
:$a \preceq_1 b$
Define $M = \struct {\N, \le}$ being an ordered set.
We will prove that:
:$M$ is a directed set.
Let $x, y \in \N$.
Thus by definition of max operation:
:$\max \set {x, y} \in \N$
Thus by definition of max operation:
:$x \le \max \set {x, y}$ and $y \le \max \set {x, y}$
{{qe... | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
:for all [[Definition:Directed Set|directed set]] $\struct {D, \precsim}$ and [[Definition:Net (Set Theory)|ne... | Let $a, b \in S$ such that
:$a \preceq_1 b$
Define $M = \struct {\N, \le}$ being an [[Definition:Ordered Set|ordered set]].
We will prove that:
:$M$ is a [[Definition:Directed Set|directed set]].
Let $x, y \in \N$.
Thus by definition of [[Definition:Max Operation|max operation]]:
:$\max \set {x, y} \in \N$
Thus by... | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Mapping_is_Increasing | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Mapping_is_Increasing | [
"Increasing Mappings",
"Limits Inferior of Nets"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Mapping",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Increasing/Mapping"
] | [
"Definition:Ordered Set",
"Definition:Directed Preordering",
"Definition:Max Operation",
"Definition:Max Operation",
"Definition:Net (Set Theory)",
"Limit Inferior of Repetition Net",
"Preceding iff Meet equals Less Operand",
"Definition:Composition of Mappings",
"Limit Inferior of Repetition Net",
... |
proofwiki-13346 | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be up-complete lattices.
Let $f: S \to T$ be a mapping such that
:for all directed set $\struct {D, \precsim}$ and net $N: D \to S$ in $S: \map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
Let $D$ be a directed subset of ... | By Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing:
:$f$ is an increasing mapping.
By Image of Directed Subset under Increasing Mapping is Directed:
:$f \sqbrk D$ is directed.
By definition of up-complete:
:$D$ and $f \sqbrk D$ admit suprema.
By Subset... | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattices]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
:for all [[Definition:Directed Set|directed set]] $\struct {D, \precsim}... | By [[Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing]]:
:$f$ is an [[Definition:Increasing Mapping|increasing mapping]].
By [[Image of Directed Subset under Increasing Mapping is Directed]]:
:$f \sqbrk D$ is [[Definition:Directed Subset|directed]].
B... | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Supremum_of_Image_is_Mapping_at_Supremum_of_Directed_Subset | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Supremum_of_Image_is_Mapping_at_Supremum_of_Directed_Subset | [
"Order Theory",
"Limits Inferior of Nets"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Mapping",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Directed Subset",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing",
"Definition:Increasing/Mapping",
"Image of Directed Subset under Increasing Mapping is Directed",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"... |
proofwiki-13347 | Prime Values of Double Factorial plus 1 | Let $n!!$ denote the double factorial function.
The sequence of positive integers $n$ such that $n!! + 1$ is prime begins:
:$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$
{{OEIS|A080778}} | We have that:
{{begin-eqn}}
{{eqn | l = 0!! + 1
| r = 1 + 1
| c = {{Defof|Double Factorial}}
}}
{{eqn | r = 2
| c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1!! + 1
| r = 1 + 1
| c = {{Defof|Double Factorial}}
}}
{{eqn | r = 2
| c = which is prime
}}
{{end-eqn}}
{{begi... | Let $n!!$ denote the [[Definition:Double Factorial|double factorial]] function.
The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] $n$ such that $n!! + 1$ is [[Definition:Prime Number|prime]] begins:
:$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502,... | We have that:
{{begin-eqn}}
{{eqn | l = 0!! + 1
| r = 1 + 1
| c = {{Defof|Double Factorial}}
}}
{{eqn | r = 2
| c = which is [[Definition:Prime Number|prime]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1!! + 1
| r = 1 + 1
| c = {{Defof|Double Factorial}}
}}
{{eqn | r = 2
| c = which... | Prime Values of Double Factorial plus 1 | https://proofwiki.org/wiki/Prime_Values_of_Double_Factorial_plus_1 | https://proofwiki.org/wiki/Prime_Values_of_Double_Factorial_plus_1 | [
"Double Factorials"
] | [
"Definition:Double Factorial",
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13348 | Limit Inferior of Restriction Net is Supremum of Image of Directed Subset | Let $L = \struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be up-complete lattices.
Let $f: S \to T$ be an increasing mapping.
Let $D \subseteq S$ be a directed subset of $S$.
Let $\struct {D, \preceq'}$ be a directed ordered subset of $L$.
Let $f \restriction D: D \to T$, the res... | We will prove that
:(lemma): $\forall j \in D: \map {\inf_L} {\paren {f \restriction D} \sqbrk {\map {\preceq'} j} } = \map f j$
Let $j \in D$.
By definitions of image of element and upper closure of element:
:$\map {\preceq'} j = j^{\succeq'}$
By Upper Closure in Ordered Subset is Intersection of Subset and Upper Clos... | Let $L = \struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattices]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $D \subseteq S$ be a [[Definition:Directed Subset... | We will prove that
:(lemma): $\forall j \in D: \map {\inf_L} {\paren {f \restriction D} \sqbrk {\map {\preceq'} j} } = \map f j$
Let $j \in D$.
By definitions of [[Definition:Image of Element under Relation|image of element]] and [[Definition:Upper Closure of Element|upper closure of element]]:
:$\map {\preceq'} j = ... | Limit Inferior of Restriction Net is Supremum of Image of Directed Subset | https://proofwiki.org/wiki/Limit_Inferior_of_Restriction_Net_is_Supremum_of_Image_of_Directed_Subset | https://proofwiki.org/wiki/Limit_Inferior_of_Restriction_Net_is_Supremum_of_Image_of_Directed_Subset | [
"Restrictions",
"Limits Inferior of Nets"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Increasing/Mapping",
"Definition:Directed Subset",
"Definition:Directed Preordering",
"Definition:Ordered Subset",
"Definition:Restriction/Mapping",
"Definition:Net (Set Theory)"
] | [
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Upper Closure/Element",
"Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure",
"Intersection is Subset",
"Image of Subset under Mapping is Subset of Image",
"Infimum of Subset",
"Infimum of Image of Upper Closure of El... |
proofwiki-13349 | Infimum of Image of Upper Closure of Element under Increasing Mapping | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered set.
Let $f: S \to T$ be an increasing mapping.
Let $x \in S$.
Then $\map \inf {f \sqbrk {x^\succeq} } = \map f x$ | By Infimum of Upper Closure of Element:
:$\inf x^\succeq = x$
By definition of infimum:
:$x$ is lower bound for $x^\succeq$
Thus by Increasing Mapping Preserves Lower Bounds:
:$\map f x$ is lower bound for $f \sqbrk {x^\succeq}$
By definition of reflexivity:
:$x \preceq x$
By definition of upper closure of element:
:$x... | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered set]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $x \in S$.
Then $\map \inf {f \sqbrk {x^\succeq} } = \map f x$ | By [[Infimum of Upper Closure of Element]]:
:$\inf x^\succeq = x$
By definition of [[Definition:Infimum of Set|infimum]]:
:$x$ is [[Definition:Lower Bound of Set|lower bound]] for $x^\succeq$
Thus by [[Increasing Mapping Preserves Lower Bounds]]:
:$\map f x$ is [[Definition:Lower Bound of Set|lower bound]] for $f \sq... | Infimum of Image of Upper Closure of Element under Increasing Mapping | https://proofwiki.org/wiki/Infimum_of_Image_of_Upper_Closure_of_Element_under_Increasing_Mapping | https://proofwiki.org/wiki/Infimum_of_Image_of_Upper_Closure_of_Element_under_Increasing_Mapping | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Increasing/Mapping"
] | [
"Infimum of Upper Closure of Element",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Increasing Mapping Preserves Lower Bounds",
"Definition:Lower Bound of Set",
"Definition:Reflexivity",
"Definition:Upper Closure/Element",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition... |
proofwiki-13350 | Poulet Numbers which are also Magic Constant for Magic Square | The sequence of Poulet numbers which are also the magic constant of a magic square begins:
:$1105, 2465, \ldots$ | From the sequence of Poulet numbers, these are Poulet numbers:
:$1105, 2465, \ldots$
Then we have:
{{begin-eqn}}
{{eqn | l = 1105
| r = \dfrac {13 \paren {13^2 + 1} } 2
| c = so $1105$ is the magic constant of the order $13$ magic square
}}
{{eqn | l = 2465
| r = \dfrac {17 \paren {17^2 + 1} } 2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Poulet Number|Poulet numbers]] which are also the [[Definition:Magic Constant|magic constant]] of a [[Definition:Magic Square|magic square]] begins:
:$1105, 2465, \ldots$ | From the [[Definition:Poulet Number/Sequence|sequence of Poulet numbers]], these are [[Definition:Poulet Number|Poulet numbers]]:
:$1105, 2465, \ldots$
Then we have:
{{begin-eqn}}
{{eqn | l = 1105
| r = \dfrac {13 \paren {13^2 + 1} } 2
| c = so $1105$ is the [[Definition:Magic Constant|magic constant]] of... | Poulet Numbers which are also Magic Constant for Magic Square | https://proofwiki.org/wiki/Poulet_Numbers_which_are_also_Magic_Constant_for_Magic_Square | https://proofwiki.org/wiki/Poulet_Numbers_which_are_also_Magic_Constant_for_Magic_Square | [
"Poulet Numbers",
"Magic Squares"
] | [
"Definition:Integer Sequence",
"Definition:Poulet Number",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square"
] | [
"Definition:Poulet Number/Sequence",
"Definition:Poulet Number",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square/Order",
"Definition:Magic Square",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square/Order",
"Definition:Magic Square",
"Definition:Carmichael Number",... |
proofwiki-13351 | Composition of Mapping and Inclusion is Restriction of Mapping | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Let $A \subseteq S$.
Then $f \circ i_A = f \restriction A$
where
:$i_A$ denotes the inclusion mapping of $A$,
:$f \restriction A$ denotes the restriction of $f$ to $A$. | By definition of inclusion mapping:
:$i_A: A \to S$
By definitions of composition of mappings and restriction of mapping:
:$f \circ i_A: A \to T$ and $f \restriction A: A \to T$
Let $a \in A$.
Thus
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ i_A} } a
| r = \map f {\map {i_A} a}
| c = {{Defof|Composition... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $A \subseteq S$.
Then $f \circ i_A = f \restriction A$
where
:$i_A$ denotes the [[Definition:Inclusion Mapping|inclusion mapping]] of $A$,
:$f \restriction A$ denotes the [[Definition:Restriction of Mapping|restriction... | By definition of [[Definition:Inclusion Mapping|inclusion mapping]]:
:$i_A: A \to S$
By definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
:$f \circ i_A: A \to T$ and $f \restriction A: A \to T$
Let $a \in A$.
Thus
{{begin-e... | Composition of Mapping and Inclusion is Restriction of Mapping | https://proofwiki.org/wiki/Composition_of_Mapping_and_Inclusion_is_Restriction_of_Mapping | https://proofwiki.org/wiki/Composition_of_Mapping_and_Inclusion_is_Restriction_of_Mapping | [
"Restrictions"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inclusion Mapping",
"Definition:Restriction/Mapping"
] | [
"Definition:Inclusion Mapping",
"Definition:Composition of Mappings",
"Definition:Restriction/Mapping"
] |
proofwiki-13352 | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.
Let $f: S \to T$ be a mapping such that
:for all directed sets $\struct {D, \precsim}$ and nets $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
Then $f$ preserves directed supr... | Let $D$ be a directed subset of $S$.
Assume that
:$D$ admits a supremum.
Thus by definition of complete lattice:
:$f \sqbrk D$ admits a supremum.
Thus by Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset:
:$\map \sup... | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
:for all [[Definition:Directed Set|directed sets]] $\struct {D, \precsim}$ and [[Definition:Net (Set Theory)... | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Assume that
:$D$ admits a [[Definition:Supremum of Set|supremum]].
Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
:$f \sqbrk D$ admits a [[Definition:Supremum of Set|supremum]].
Thus by [[Mapping at Limit Inferior Precedes L... | Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Mapping_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_Mapping_and_Sequence_implies_Mapping_Preserves_Directed_Suprema | [
"Order Theory",
"Limits Inferior of Nets"
] | [
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Complete Lattice",
"Definition:Supremum of Set",
"Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset"
] |
proofwiki-13353 | Thurston's Geometrization Conjecture | When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:
:$(1): \quad$ Euclidean geometry
:$(2): \quad$ Hyperbolic geometry
:$(3): \quad$ Spherical geometry
:$(4): \q... | {{ProofWanted|Discuss {{AuthorRef|Grigori Perelman}}'s work -- it is generally believed his proof of this is valid, in which case this needs to be renamed to Theorem.}}
{{Namedfor|William Paul Thurston|cat = Thurston}} | When a [[Definition:Topological Manifold|topological manifold]] of [[Definition:Dimension of Topological Manifold|dimension $3$]] has been split into its [[Definition:Connected Sum|connected sum]] and the [[Definition:Jaco-Shalen-Johannson Torus Decomposition|Jaco-Shalen-Johannson torus decomposition]], the remaining [... | {{ProofWanted|Discuss {{AuthorRef|Grigori Perelman}}'s work -- it is generally believed his proof of this is valid, in which case this needs to be renamed to Theorem.}}
{{Namedfor|William Paul Thurston|cat = Thurston}} | Thurston's Geometrization Conjecture | https://proofwiki.org/wiki/Thurston's_Geometrization_Conjecture | https://proofwiki.org/wiki/Thurston's_Geometrization_Conjecture | [
"Topological Manifolds"
] | [
"Definition:Topological Manifold",
"Definition:Dimension (Topology)/Topological Manifold",
"Definition:Connected Sum",
"Definition:Jaco-Shalen-Johannson Torus Decomposition",
"Definition:Component of Decomposed Manifold",
"Definition:Geometry of Manifold",
"Definition:Euclidean Geometry",
"Definition:... | [] |
proofwiki-13354 | Mapping Preserves Directed Suprema implies Mapping is Continuous | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be up-complete ordered sets with Scott topologies.
Let $f: S \to T$ be a directed suprema preserving mapping.
Then $f$ is continuous. | Let $P$ be a closed subset of $T$.
By Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set:
:$P$ is lower and closed under directed suprema.
We will prove that
:for all directed subset $D$ of $S$: $D \subseteq f^{-1} \sqbrk P \implies \sup D \in f^{-1} \sqbrk P$
Let $D$ be a directed ... | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered sets]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definition:Mapping Preserves Supremum/Directed|directed suprema preserving]] [[Definit... | Let $P$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Subset|subset]] of $T$.
By [[Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set]]:
:$P$ is [[Definition:Lower Section|lower]] and [[Definition:Closed under Directed Suprema|closed under directed suprema]].
We wi... | Mapping Preserves Directed Suprema implies Mapping is Continuous | https://proofwiki.org/wiki/Mapping_Preserves_Directed_Suprema_implies_Mapping_is_Continuous | https://proofwiki.org/wiki/Mapping_Preserves_Directed_Suprema_implies_Mapping_is_Continuous | [
"Topological Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Scott Topology",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Closed Set/Topology",
"Definition:Subset",
"Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set",
"Definition:Lower Section",
"Definition:Closed under Directed Suprema",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Mapping Pres... |
proofwiki-13355 | Secant of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sec \paren {a + b i} = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}$
where:
:$\sec$ denotes the complex secant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sin... | {{begin-eqn}}
{{eqn | l = \sec \paren {a + b i}
| r = \frac 1 {\cos \paren {a + b i} }
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \dfrac 1 {\cos a \cosh b - i \sin a \sinh b}
| c = Cosine of Complex Number
}}
{{eqn | r = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\paren {\cos a \cosh b + ... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sec \paren {a + b i} = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}$
where:
:$\sec$ denotes the [[Definition:Complex Secant Function|complex secan... | {{begin-eqn}}
{{eqn | l = \sec \paren {a + b i}
| r = \frac 1 {\cos \paren {a + b i} }
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \dfrac 1 {\cos a \cosh b - i \sin a \sinh b}
| c = [[Cosine of Complex Number]]
}}
{{eqn | r = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\paren {\cos a \cosh ... | Secant of Complex Number | https://proofwiki.org/wiki/Secant_of_Complex_Number | https://proofwiki.org/wiki/Secant_of_Complex_Number | [
"Secant Function",
"Complex Numbers"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Secant Function/Complex",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Cosine of Complex Number",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-13356 | Tangent of Complex Number/Formulation 1 | :$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \frac {\sin \paren {a + b i} } {\cos \paren {a + b i} }
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = Sine of Complex Number and Cosine of Complex Number... | :$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \frac {\sin \paren {a + b i} } {\cos \paren {a + b i} }
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = [[Sine of Complex Number]] and [[Cosine of Complex ... | Tangent of Complex Number/Formulation 1 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_1 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_1 | [
"Tangent of Complex Number"
] | [] | [
"Sine of Complex Number",
"Cosine of Complex Number"
] |
proofwiki-13357 | Tangent of Complex Number/Formulation 2 | :$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = Tangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\tan a \cosh b + i \sinh b} {\cosh b - i \tan a \sinh b}
| c = multiplying denominator and numerato... | :$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = [[Tangent of Complex Number/Formulation 1|Tangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\tan a \cosh b + i \sinh b} {\cosh b - i \tan a \sinh b}
... | Tangent of Complex Number/Formulation 2 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_2 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_2 | [
"Tangent of Complex Number"
] | [] | [
"Tangent of Complex Number/Formulation 1",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-13358 | Tangent of Complex Number/Formulation 3 | :$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \frac {\tan a + i \tanh b} {1 - i \tan a \tanh b}
| c = Tangent of Complex Number: Formulation 2
}}
{{eqn | r = \frac {\paren {\tan a + i \tanh b} \paren {1 + i \tan a \tanh b} } {1 + \tan ^2 a \tanh ^2 b}
| c = multiplying denominator and numerato... | :$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \frac {\tan a + i \tanh b} {1 - i \tan a \tanh b}
| c = [[Tangent of Complex Number/Formulation 2|Tangent of Complex Number: Formulation 2]]
}}
{{eqn | r = \frac {\paren {\tan a + i \tanh b} \paren {1 + i \tan a \tanh b} } {1 + \tan ^2 a \tanh ^2 b}
... | Tangent of Complex Number/Formulation 3 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_3 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_3 | [
"Tangent of Complex Number"
] | [] | [
"Tangent of Complex Number/Formulation 2",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-13359 | Cosecant of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$
where:
:$\csc$ denotes the complex cosecant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\s... | {{begin-eqn}}
{{eqn | l = \csc \paren {a + b i}
| r = \frac 1 {\sin \paren {a + b i} }
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \dfrac 1 {\sin a \cosh b + i \cos a \sinh b}
| c = Sine of Complex Number
}}
{{eqn | r = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\paren {\sin a \cosh b - ... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$
where:
:$\csc$ denotes the [[Definition:Complex Cosecant Function|complex cos... | {{begin-eqn}}
{{eqn | l = \csc \paren {a + b i}
| r = \frac 1 {\sin \paren {a + b i} }
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \dfrac 1 {\sin a \cosh b + i \cos a \sinh b}
| c = [[Sine of Complex Number]]
}}
{{eqn | r = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\paren {\sin a \cosh ... | Cosecant of Complex Number | https://proofwiki.org/wiki/Cosecant_of_Complex_Number | https://proofwiki.org/wiki/Cosecant_of_Complex_Number | [
"Cosecant Function",
"Complex Numbers"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosecant/Complex Function",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Sine of Complex Number",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-13360 | Cotangent of Complex Number/Formulation 1 | :$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$ | {{begin-eqn}}
{{eqn | l = \cot \paren {a + b i}
| r = \frac {\cos \paren {a + b i} } {\sin \paren {a + b i} }
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
| c = Sine of Complex Number and Cosine of Complex Numb... | :$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$ | {{begin-eqn}}
{{eqn | l = \cot \paren {a + b i}
| r = \frac {\cos \paren {a + b i} } {\sin \paren {a + b i} }
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
| c = [[Sine of Complex Number]] and [[Cosine of Comple... | Cotangent of Complex Number/Formulation 1 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_1 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_1 | [
"Cotangent of Complex Number"
] | [] | [
"Sine of Complex Number",
"Cosine of Complex Number"
] |
proofwiki-13361 | Cotangent of Complex Number/Formulation 2 | :$\map \cot {a + b i} = \dfrac {-1 - i \cot a \coth b} {\cot a - i \coth b}$ | {{begin-eqn}}
{{eqn | l = \map \cot {a + b i}
| r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
| c = Cotangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\cot a \cosh b - i \sinh b} {\cosh b + i \cot a \sinh b}
| c = multiplying denominator and numerato... | :$\map \cot {a + b i} = \dfrac {-1 - i \cot a \coth b} {\cot a - i \coth b}$ | {{begin-eqn}}
{{eqn | l = \map \cot {a + b i}
| r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
| c = [[Cotangent of Complex Number/Formulation 1|Cotangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\cot a \cosh b - i \sinh b} {\cosh b + i \cot a \sinh b}
... | Cotangent of Complex Number/Formulation 2 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_2 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_2 | [
"Cotangent of Complex Number"
] | [] | [
"Cotangent of Complex Number/Formulation 1",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-13362 | Cotangent of Complex Number/Formulation 3 | :$\map \cot {a + b i} = \dfrac {\cot a \coth^2 b - \cot a} {\cot^2 a + \coth^2 b} + \dfrac {-\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$ | {{begin-eqn}}
{{eqn | l = \map \cot {a + b i}
| r = \dfrac {1 + i \cot a \coth b} {\cot a - i \coth b}
| c = Cotangent of Complex Number: Formulation 2
}}
{{eqn | r = \dfrac {\paren {1 + i \cot a \coth b} \paren {\cot a + i \coth b} } {\paren {\cot a - i \coth b} \paren {\cot a + i \coth b} }
| c = mu... | :$\map \cot {a + b i} = \dfrac {\cot a \coth^2 b - \cot a} {\cot^2 a + \coth^2 b} + \dfrac {-\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$ | {{begin-eqn}}
{{eqn | l = \map \cot {a + b i}
| r = \dfrac {1 + i \cot a \coth b} {\cot a - i \coth b}
| c = [[Cotangent of Complex Number/Formulation 2|Cotangent of Complex Number: Formulation 2]]
}}
{{eqn | r = \dfrac {\paren {1 + i \cot a \coth b} \paren {\cot a + i \coth b} } {\paren {\cot a - i \coth b... | Cotangent of Complex Number/Formulation 3 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_3 | https://proofwiki.org/wiki/Cotangent_of_Complex_Number/Formulation_3 | [
"Cotangent of Complex Number"
] | [] | [
"Cotangent of Complex Number/Formulation 2",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-13363 | Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous {{iff}}:
:for all directed set $\struct {D, \precsim}$ and net $N: D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ ... | === Sufficient Condition ===
Assume that
:$f$ is continuous.
Let $\struct {D, \precsim}$ be a directed set.
Let $N: D \to S$ be a net in $S$.
{{AimForCont}}
:$\map f {\liminf N} \npreceq_2 \map \liminf {f \circ N}$
By definition of lower closure of element:
:$\map f {\liminf N} \notin \paren {\map \liminf {f \circ N} }... | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then $f$ is [[Definition:Continuous ... | === Sufficient Condition ===
Assume that
:$f$ is [[Definition:Continuous (Topology)|continuous]].
Let $\struct {D, \precsim}$ be a [[Definition:Directed Set|directed set]].
Let $N: D \to S$ be a [[Definition:Net (Set Theory)|net]] in $S$.
{{AimForCont}}
:$\map f {\liminf N} \npreceq_2 \map \liminf {f \circ N}$
By ... | Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_of_Mapping_and_Sequence | https://proofwiki.org/wiki/Continuous_iff_Mapping_at_Limit_Inferior_Precedes_Limit_Inferior_of_Composition_of_Mapping_and_Sequence | [
"Topological Order Theory",
"Limits Inferior of Nets"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)"
] | [
"Definition:Continuous Mapping (Topology)",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Lower Closure/Element",
"Definition:Relative Complement",
"Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set",
"Definition:Closed Set/Topology",
"De... |
proofwiki-13364 | Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.
Let $f: S \to T$ be an increasing mapping.
Let $\struct {D, \precsim}$ be a directed set.
Let $N: D \to S$ be a net in $S$.
Let $j \in D$.
Then $\map f {\map \inf {N \sqbrk {\map \precsim j} } } \preceq_2... | By definitions of image of set and composition of mappings:
:$f \sqbrk {N \sqbrk {\map \precsim j} } = \paren {f \circ N} \sqbrk {\map \precsim j}$
By definition of complete lattice:
:$f \sqbrk {N \sqbrk {\map \precsim j} }$ and $N \sqbrk {\map \precsim j}$ admit infima.
Thus by Subset and Image Admit Infima and Mappin... | Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
Let $\struct {D, \precsim}$ be a [[Definition:Directed Set|directed set]].
Let $N: D \to S$ ... | By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Composition of Mappings|composition of mappings]]:
:$f \sqbrk {N \sqbrk {\map \precsim j} } = \paren {f \circ N} \sqbrk {\map \precsim j}$
By definition of [[Definition:Complete Lattice|complete lattice]]:
:$f \sqbrk {N \sqbrk... | Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence | https://proofwiki.org/wiki/Mapping_is_Increasing_implies_Mapping_at_Infimum_for_Sequence_Precedes_Infimum_for_Composition_of_Mapping_and_Sequence | https://proofwiki.org/wiki/Mapping_is_Increasing_implies_Mapping_at_Infimum_for_Sequence_Precedes_Infimum_for_Composition_of_Mapping_and_Sequence | [
"Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Increasing/Mapping",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Composition of Mappings",
"Definition:Complete Lattice",
"Definition:Infimum of Set",
"Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum"
] |
proofwiki-13365 | Set of Infima for Sequence is Directed | Let $\struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $\struct {A, \precsim}$ be a non-empty directed set.
Let $Z: A \to S$ be a net.
Let $D = \set {\map \inf {Z \sqbrk {\map \precsim j} }: j \in A}$ be a subset of $S$.
Then $D$ is directed. | By definition of non-empty set:
:$\exists j: j \in A$
By definition of $D$:
:$\inf \left({Z \sqbrk {\map \precsim j} }\right) \in D$
Hence by definition:
:$D$ is a non-empty set.
Let $x, y \in D$.
By definition of $D$:
:$\exists j_1 \in A: x = \map \inf {Z \sqbrk {\map \precsim {j_1} } }$
and
:$\exists j_2 \in A: y = \... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $\struct {A, \precsim}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Directed Set|directed set]].
Let $Z: A \to S$ be a [[Definition:Net (Set Theory)|net]].
Let $D = \set {\map \inf {Z \sqbrk {\map \precsi... | By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists j: j \in A$
By definition of $D$:
:$\inf \left({Z \sqbrk {\map \precsim j} }\right) \in D$
Hence by definition:
:$D$ is a [[Definition:Non-Empty Set|non-empty set]].
Let $x, y \in D$.
By definition of $D$:
:$\exists j_1 \in A: x = \map \inf {Z \... | Set of Infima for Sequence is Directed | https://proofwiki.org/wiki/Set_of_Infima_for_Sequence_is_Directed | https://proofwiki.org/wiki/Set_of_Infima_for_Sequence_is_Directed | [
"Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Non-Empty Set",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Subset",
"Definition:Directed Subset"
] | [
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Directed Preordering",
"Preceding implies Image is Subset of Image",
"Image of Subset under Mapping is Subset of Image",
"Infimum of Subset"
] |
proofwiki-13366 | Preceding implies Image is Subset of Image | Let $\struct {S, \precsim}$ be a preordered set.
Let $x, y \in S$ such that
:$x \precsim y$
Then $\map \precsim y \subseteq \mathord {\map \precsim x}$
where $\map \precsim y$ denotes the image of $y$ under $\precsim$. | Let $z \in \mathord {\map \precsim y}$
By definition of image of element:
:$y \precsim z$
By definition of transitivity:
:$x \precsim z$
Thus by definition of image of element:
:$z \in \mathord {\map \precsim x}$
{{qed}} | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $x, y \in S$ such that
:$x \precsim y$
Then $\map \precsim y \subseteq \mathord {\map \precsim x}$
where $\map \precsim y$ denotes the [[Definition:Image of Element under Relation|image]] of $y$ under $\precsim$. | Let $z \in \mathord {\map \precsim y}$
By definition of [[Definition:Image of Element under Relation|image of element]]:
:$y \precsim z$
By definition of [[Definition:Transitivity|transitivity]]:
:$x \precsim z$
Thus by definition of [[Definition:Image of Element under Relation|image of element]]:
:$z \in \mathord {... | Preceding implies Image is Subset of Image | https://proofwiki.org/wiki/Preceding_implies_Image_is_Subset_of_Image | https://proofwiki.org/wiki/Preceding_implies_Image_is_Subset_of_Image | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Image (Set Theory)/Relation/Element"
] | [
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Transitive",
"Definition:Image (Set Theory)/Relation/Element"
] |
proofwiki-13367 | Universal Property of Group Ring | Let $R$ be a commutative ring with unity.
Let $G$ be a group.
Let $R \sqbrk G$ be the corresponding group ring.
Let $S$ be a commutative ring with unity.
Let $\phi: R \to S$ be a ring homomorphism.
Let $\beta : G \to S^\times$ be a group homomorphism, where $S^\times$ is the multiplicative group of $S$.
Then there exis... | {{ProofWanted}}
Category:Universal Properties
Category:Group Rings
67dm0aljrwzkn75c8os68bnp1hssab4 | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $G$ be a [[Definition:Group|group]].
Let $R \sqbrk G$ be the corresponding [[Definition:Group Ring|group ring]].
Let $S$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\phi: R \to S$ be a [[De... | {{ProofWanted}}
[[Category:Universal Properties]]
[[Category:Group Rings]]
67dm0aljrwzkn75c8os68bnp1hssab4 | Universal Property of Group Ring | https://proofwiki.org/wiki/Universal_Property_of_Group_Ring | https://proofwiki.org/wiki/Universal_Property_of_Group_Ring | [
"Universal Properties",
"Group Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Group",
"Definition:Group Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Ring Homomorphism",
"Definition:Group Homomorphism",
"Definition:Multiplicative Group of Ring",
"Definition:Ring Homomorphism"
] | [
"Category:Universal Properties",
"Category:Group Rings"
] |
proofwiki-13368 | Continuous iff Directed Suprema Preserving | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous {{iff}} $f$ preserves directed suprema. | === Sufficient Condition ===
Assume that
:$f$ is continuous.
By Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence:
:for all directed set $\struct {D, \precsim}$ and net $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
Thus by Mapping at Li... | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then $f$ is [[Definition:Continuous ... | === Sufficient Condition ===
Assume that
:$f$ is [[Definition:Continuous (Topology)|continuous]].
By [[Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence]]:
:for all [[Definition:Directed Set|directed set]] $\struct {D, \precsim}$ and [[Definition:Net (Set Theory)|... | Continuous iff Directed Suprema Preserving | https://proofwiki.org/wiki/Continuous_iff_Directed_Suprema_Preserving | https://proofwiki.org/wiki/Continuous_iff_Directed_Suprema_Preserving | [
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Definition:Continuous Mapping (Topology)",
"Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence impl... |
proofwiki-13369 | Universal Property of Quotient of Topological Group | Let $G$ and $H$ be topological groups.
Let $N$ be a normal subgroup of $G$.
Let $\pi : G \to G/N$ be the quotient mapping.
Let $f : G \to H$ be a continuous group homomorphism whose kernel contains $N$.
Then there exists a unique continuous group homomorphism $\overline f: G / N \to H$ such that:
:$f = \overline f \cir... | Because $N \subset \ker f$, $f$ is constant on the cosets of $N$.
Thus $f$ is invariant under congruence modulo $N$.
By Universal Property of Quotient Set, there exists a unique mapping $\overline f: G / N \to H$ such that:
:$f = \overline f \circ \pi$
It suffices to verify that it is a continuous group homomorphism.
B... | Let $G$ and $H$ be [[Definition:Topological Group|topological groups]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $\pi : G \to G/N$ be the [[Definition:Quotient Mapping|quotient mapping]].
Let $f : G \to H$ be a [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Group ... | Because $N \subset \ker f$, $f$ is constant on the [[Definition:Coset|cosets]] of $N$.
Thus $f$ is [[Definition:Invariant Mapping Under Equivalence Relation|invariant under]] [[Definition:Congruence Modulo Subgroup|congruence modulo]] $N$.
By [[Universal Property of Quotient Set]], there exists a [[Definition:Unique ... | Universal Property of Quotient of Topological Group | https://proofwiki.org/wiki/Universal_Property_of_Quotient_of_Topological_Group | https://proofwiki.org/wiki/Universal_Property_of_Quotient_of_Topological_Group | [
"Topological Groups",
"Universal Properties"
] | [
"Definition:Topological Group",
"Definition:Normal Subgroup",
"Definition:Quotient Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Unique up to Isomorphism",
"Definition:Continuous Mapping (Topology)",
"Def... | [
"Definition:Coset",
"Definition:Invariant Mapping Under Equivalence Relation",
"Definition:Congruence Modulo Subgroup",
"Universal Property of Quotient Set",
"Definition:Unique up to Isomorphism",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Group Homomorphism",
"Uni... |
proofwiki-13370 | Universal Property of Quotient Set | Let $X$ and $Y$ be sets.
Let $\sim$ be an equivalence relation on $X$.
Let $\pi: X \to X / {\sim}$ be the quotient mapping to the quotient set.
Let $f: X \to Y$ be $\sim$-invariant.
Then there exists a unique mapping $\overline f : X / {\sim} \to Y$ such that $f = \overline f \circ \pi$.
:<nowiki>$\xymatrix{
X \ar[d]_\... | === Existence ===
For $x \in X$, let $\eqclass x {} \in X / {\sim}$ denote its equivalence class under $\sim$.
Define the relation $\overline f: X / {\sim} \to Y$ by:
:$\tuple {\eqclass x {}, y} \in \overline f \iff \map f x = y$
We check that this is a well-defined mapping.
By Quotient Mapping is Surjection, $f$ is le... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
Let $\pi: X \to X / {\sim}$ be the [[Definition:Quotient Mapping|quotient mapping]] to the [[Definition:Quotient Set|quotient set]].
Let $f: X \to Y$ be [[Definition:Invariant Mapping Under E... | === Existence ===
For $x \in X$, let $\eqclass x {} \in X / {\sim}$ denote its [[Definition:Equivalence Class|equivalence class]] under $\sim$.
Define the [[Definition:Relation|relation]] $\overline f: X / {\sim} \to Y$ by:
:$\tuple {\eqclass x {}, y} \in \overline f \iff \map f x = y$
We check that this is a [[Defi... | Universal Property of Quotient Set | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Set | https://proofwiki.org/wiki/Universal_Property_of_Quotient_Set | [
"Equivalence Relations",
"Universal Properties"
] | [
"Definition:Set",
"Definition:Equivalence Relation",
"Definition:Quotient Mapping",
"Definition:Quotient Set",
"Definition:Invariant Mapping Under Equivalence Relation",
"Definition:Mapping"
] | [
"Definition:Equivalence Class",
"Definition:Relation",
"Definition:Well-Defined/Mapping",
"Quotient Mapping is Surjection",
"Definition:Left-Total Relation",
"Definition:Equivalence Class",
"Definition:Invariant Mapping Under Equivalence Relation",
"Definition:Many-to-One Relation",
"Definition:Well... |
proofwiki-13371 | Continuous iff Way Below iff There Exists Element that Way Below and Way Below | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete continuous topological lattices with Scott topologies.
Let $f: S \to T$ be a mapping.
Then $f$ is continuous {{iff}}
:$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$
{{explain|link to defi... | === Sufficient Condition ===
Assume that
:$f$ is continuous.
By Continuous iff Directed Suprema Preserving:
:$f$ preserves directed suprema.
By Directed Suprema Preserving Mapping at Element is Supremum:
:$\forall x \in S: \map f x = \sup \set {\map f w: w \in S \land w \ll x}$
Thus by Mapping at Element is Supremum im... | Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
Let $f: S \to T$ be a [[Definition:Mapping|... | === Sufficient Condition ===
Assume that
:$f$ is [[Definition:Continuous (Topology)|continuous]].
By [[Continuous iff Directed Suprema Preserving]]:
:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
By [[Directed Suprema Preserving Mapping at Element is Supremum]]:
:$\forall x \in S... | Continuous iff Way Below iff There Exists Element that Way Below and Way Below | https://proofwiki.org/wiki/Continuous_iff_Way_Below_iff_There_Exists_Element_that_Way_Below_and_Way_Below | https://proofwiki.org/wiki/Continuous_iff_Way_Below_iff_There_Exists_Element_that_Way_Below_and_Way_Below | [
"Topological Order Theory",
"Way Below Relation",
"Continuous Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Continuous Ordered Set",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Continuous Mapping (Topology)",
"Continuous iff Directed Suprema Preserving",
"Definition:Mapping Preserves Supremum/Directed",
"Directed Suprema Preserving Mapping at Element is Supremum",
"Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below",
... |
proofwiki-13372 | Finite Symmetric Group is Ambivalent | Let $n$ be a natural number.
Let $S_n$ be a symmetric group of order $n$.
Then $S_n$ is ambivalent. | The Conjugacy Classes of Symmetric Group are determined uniquely by the cycle type of the elements.
Since any element in $S_n$ is of the same cycle type with its inverse, they are in the same conjugacy class.
Hence they are conjugates of each other.
This implies that $S_n$ is ambivalent.
{{qed}}
Category:Ambivalent Gro... | Let $n$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ be a [[Definition:Symmetric Group|symmetric group]] of [[Definition:Order of Structure|order $n$]].
Then $S_n$ is [[Definition:Ambivalent Group|ambivalent]]. | The [[Conjugacy Classes of Symmetric Group]] are determined uniquely by the [[Definition:Cycle Type|cycle type]] of the [[Definition:Element|elements]].
Since any element in $S_n$ is of the same [[Definition:Cycle Type|cycle type]] with its [[Definition:Inverse Element|inverse]], they are in the same [[Definition:Conj... | Finite Symmetric Group is Ambivalent | https://proofwiki.org/wiki/Finite_Symmetric_Group_is_Ambivalent | https://proofwiki.org/wiki/Finite_Symmetric_Group_is_Ambivalent | [
"Ambivalent Groups"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Ambivalent Group"
] | [
"Conjugacy Classes of Symmetric Group",
"Definition:Cycle Type",
"Definition:Element",
"Definition:Cycle Type",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Conjugacy Class",
"Definition:Conjugate (Group Theory)/Element",
"Definition:Ambivalent Group",
"Category:Ambivalent Groups"
] |
proofwiki-13373 | Alternating Groups that are Ambivalent | Let $n$ be a natural number.
Then the $n$th alternating group $A_n$ is ambivalent {{iff}} $n \in \set {1, 2, 5, 6, 10, 14}$.
{{OEIS|A115200}} | {{ProofWanted|Seems related to the representation theory of alternating group.}}
Category:Alternating Groups
Category:Ambivalent Groups
ewrv3cjn4p7ejdr540kmndavpsgvojt | Let $n$ be a [[Definition:Natural Number|natural number]].
Then the $n$th [[Definition:Alternating Group|alternating group]] $A_n$ is [[Definition:Ambivalent Group|ambivalent]] {{iff}} $n \in \set {1, 2, 5, 6, 10, 14}$.
{{OEIS|A115200}} | {{ProofWanted|Seems related to the representation theory of alternating group.}}
[[Category:Alternating Groups]]
[[Category:Ambivalent Groups]]
ewrv3cjn4p7ejdr540kmndavpsgvojt | Alternating Groups that are Ambivalent | https://proofwiki.org/wiki/Alternating_Groups_that_are_Ambivalent | https://proofwiki.org/wiki/Alternating_Groups_that_are_Ambivalent | [
"Alternating Groups",
"Ambivalent Groups"
] | [
"Definition:Natural Numbers",
"Definition:Alternating Group",
"Definition:Ambivalent Group"
] | [
"Category:Alternating Groups",
"Category:Ambivalent Groups"
] |
proofwiki-13374 | Transitivity of Big-O Estimates/General | Let $X$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.
Let $f, g, h: X \to V$ be functions.
Let $x_0 \in X$.
Let $f = \map \OO g$ and $g = \map \OO h$ as $x \to x_0$, where $\OO$ denotes big-$\OO$ notation.
Then $f = \map \OO h$ as $x \to x_0$. | Because $f = \map \OO g$ and $g = \map \OO h$, there exist neighborhoods $U$ and $V$ of $x_0$ and real numbers $c, d \ge 0$ such that:
:$\norm {\map f x} \le c \cdot \norm {\map g x}$ for all $x \in U$
:$\norm {\map g x} \le d \cdot \norm {\map h x}$ for all $x \in V$.
By Intersection of Neighborhoods in Topological Sp... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$.
Let $f, g, h: X \to V$ be functions.
Let $x_0 \in X$.
Let $f = \map \OO g$ and $g = \map \OO h$ as... | Because $f = \map \OO g$ and $g = \map \OO h$, there exist [[Definition:Neighborhood of Point in Topological Space|neighborhoods]] $U$ and $V$ of $x_0$ and [[Definition:Real Number|real numbers]] $c, d \ge 0$ such that:
:$\norm {\map f x} \le c \cdot \norm {\map g x}$ for all $x \in U$
:$\norm {\map g x} \le d \cdot \n... | Transitivity of Big-O Estimates/General | https://proofwiki.org/wiki/Transitivity_of_Big-O_Estimates/General | https://proofwiki.org/wiki/Transitivity_of_Big-O_Estimates/General | [
"Big-O Notation"
] | [
"Definition:Topological Space",
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Big-O Notation"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Real Number",
"Intersection of Neighborhoods in Topological Space is Neighborhood",
"Definition:Neighborhood (Topology)/Point",
"Category:Big-O Notation"
] |
proofwiki-13375 | Transitivity of Big-O Estimates/Sequences | Let $\sequence {a_n}$, $\sequence {b_n}$ and $\sequence {c_n}$ be sequences of real or complex numbers.
Let $a_n = \map \OO {\sequence {b_n} }$ and $b_n = \map \OO {\sequence {c_n} }$, where $\OO$ denotes big-$\OO$ notation.
Then $a_n = \map \OO {\sequence {c_n} }$. | Because $a_n = \map \OO {\sequence {b_n} }$, there exists $K \ge 0$ and $n_0 \in \N$ such that $\size {a_n} \le K \cdot \size {b_n}$ for $n \ge n_0$.
Because $b_n = \map \OO {\sequence {c_n} }$, there exists $L \ge 0$ and $n_1 \in \N$ such that $\size {b_n} \le L \cdot \size {c_n}$ for $n \ge n_1$.
Then $\size {a_n} \l... | Let $\sequence {a_n}$, $\sequence {b_n}$ and $\sequence {c_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let $a_n = \map \OO {\sequence {b_n} }$ and $b_n = \map \OO {\sequence {c_n} }$, where $\OO$ denotes [[Definition:Big-O Notation for S... | Because $a_n = \map \OO {\sequence {b_n} }$, there exists $K \ge 0$ and $n_0 \in \N$ such that $\size {a_n} \le K \cdot \size {b_n}$ for $n \ge n_0$.
Because $b_n = \map \OO {\sequence {c_n} }$, there exists $L \ge 0$ and $n_1 \in \N$ such that $\size {b_n} \le L \cdot \size {c_n}$ for $n \ge n_1$.
Then $\size {a_n} ... | Transitivity of Big-O Estimates/Sequences | https://proofwiki.org/wiki/Transitivity_of_Big-O_Estimates/Sequences | https://proofwiki.org/wiki/Transitivity_of_Big-O_Estimates/Sequences | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Big-O Notation/Sequence"
] | [
"Category:Big-O Notation"
] |
proofwiki-13376 | Conjugate of Cycle | Let $n \ge 1$ be a natural number.
Let $S_n$ be the symmetric group on $n$ letters.
Let $\pi, \sigma \in S_n$.
Let $\sigma$ be a cycle of length $k$.
Then the conjugate $\pi \sigma \pi^{-1}$ is a cycle of length $k$. | Follows directly from Conjugate Permutations have Same Cycle Type.
{{qed}}
Category:Cyclic Permutations
Category:Conjugacy
32y4tmmpdspvu6gfxihl1bscksbwenj | Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ be the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\pi, \sigma \in S_n$.
Let $\sigma$ be a [[Definition:Cyclic Permutation|cycle]] of [[Definition:Length of Cyclic Permutation|length]] $k$.
Then the [[De... | Follows directly from [[Conjugate Permutations have Same Cycle Type]].
{{qed}}
[[Category:Cyclic Permutations]]
[[Category:Conjugacy]]
32y4tmmpdspvu6gfxihl1bscksbwenj | Conjugate of Cycle | https://proofwiki.org/wiki/Conjugate_of_Cycle | https://proofwiki.org/wiki/Conjugate_of_Cycle | [
"Cyclic Permutations",
"Conjugacy"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Cyclic Permutation",
"Definition:Cyclic Permutation",
"Definition:Conjugate (Group Theory)/Element",
"Definition:Cyclic Permutation",
"Definition:Cyclic Permutation"
] | [
"Conjugate Permutations have Same Cycle Type",
"Category:Cyclic Permutations",
"Category:Conjugacy"
] |
proofwiki-13377 | Center of Group is Kernel of Conjugacy Action | Let $G$ be a group.
Let $Z$ be the kernel of the conjugacy action.
Then $Z$ is the center of $G$. | {{begin-eqn}}
{{eqn | o =
| r = x \text { is in the kernel of the conjugacy action}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in G
| o =
| r = x y x^{-1} = y
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in G
| o =
| r = x y = y x
| c = Division Laws for Groups
}}... | Let $G$ be a [[Definition:Group|group]].
Let $Z$ be the [[Definition:Kernel of Group Action|kernel]] of the [[Definition:Conjugacy Action|conjugacy action]].
Then $Z$ is the [[Definition:Center of Group|center]] of $G$. | {{begin-eqn}}
{{eqn | o =
| r = x \text { is in the kernel of the conjugacy action}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in G
| o =
| r = x y x^{-1} = y
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y \in G
| o =
| r = x y = y x
| c = [[Division Laws for Groups]... | Center of Group is Kernel of Conjugacy Action | https://proofwiki.org/wiki/Center_of_Group_is_Kernel_of_Conjugacy_Action | https://proofwiki.org/wiki/Center_of_Group_is_Kernel_of_Conjugacy_Action | [
"Conjugacy Action",
"Centers of Groups"
] | [
"Definition:Group",
"Definition:Kernel of Group Action",
"Definition:Conjugacy Action",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Division Laws for Groups",
"Category:Conjugacy Action",
"Category:Centers of Groups"
] |
proofwiki-13378 | Little-O Implies Big-O/General Result | Let $X$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$
Let $f, g: X \to V$ be mappings.
Let $x_0 \in X$.
Let $f = \map \oo g$ as $x \to x_0$, where $\oo$ denotes little-$\oo$ notation.
Then $f = \map \OO g$ as $x \to x_0$, where $\OO$ denotes big-$\OO$ notation. | From the definition of little-$\oo$ notation:
:there exists a neighborhood $U$ of $x_0$ such that $\norm {\map f x} \le \norm {\map g x}$ for all $x \in U$.
By definition of big-$\OO$ notation, $f = \map \OO g$ as $x \to x_0$.
{{qed}}
Category:Big-O Notation
Category:Little-O Notation
krsegn7o79z9ixc9c2z4qg0elkoriej | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$
Let $f, g: X \to V$ be [[Definition:Mapping|mappings]].
Let $x_0 \in X$.
Let $f = \map \oo g$ as $x... | From the definition of [[Definition:Little-O Notation|little-$\oo$ notation]]:
:there exists a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] $U$ of $x_0$ such that $\norm {\map f x} \le \norm {\map g x}$ for all $x \in U$.
By definition of [[Definition:Big-O Notation|big-$\OO$ notation]], $f =... | Little-O Implies Big-O/General Result | https://proofwiki.org/wiki/Little-O_Implies_Big-O/General_Result | https://proofwiki.org/wiki/Little-O_Implies_Big-O/General_Result | [
"Big-O Notation",
"Little-O Notation"
] | [
"Definition:Topological Space",
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Mapping",
"Definition:Little-O Notation",
"Definition:Big-O Notation"
] | [
"Definition:Little-O Notation",
"Definition:Neighborhood (Topology)/Point",
"Definition:Big-O Notation",
"Category:Big-O Notation",
"Category:Little-O Notation"
] |
proofwiki-13379 | Little-O Implies Big-O/Sequences | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
Let $a_n = \map \oo {b_n}$ where $\oo$ denotes little-$\oo$ notation.
Then $a_n = \map \OO {b_n}$ where $\OO$ denotes big-$\OO$ notation. | Because $a_n = \map \oo {b_n}$, there exists $n_0 \in \N$ such that $\size {a_n} \le 1 \cdot \size {b_n}$ for $n \ge n_0$.
Thus $a_n = \map \OO {b_n}$.
{{qed}}
Category:Big-O Notation
Category:Little-O Notation
m5v6mqpzbxxgth2n45of1yj7l1r0ani | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let $a_n = \map \oo {b_n}$ where $\oo$ denotes [[Definition:Little-O Notation|little-$\oo$ notation]].
Then $a_n = \map \OO {b_n}$ where $\OO$ denotes ... | Because $a_n = \map \oo {b_n}$, there exists $n_0 \in \N$ such that $\size {a_n} \le 1 \cdot \size {b_n}$ for $n \ge n_0$.
Thus $a_n = \map \OO {b_n}$.
{{qed}}
[[Category:Big-O Notation]]
[[Category:Little-O Notation]]
m5v6mqpzbxxgth2n45of1yj7l1r0ani | Little-O Implies Big-O/Sequences | https://proofwiki.org/wiki/Little-O_Implies_Big-O/Sequences | https://proofwiki.org/wiki/Little-O_Implies_Big-O/Sequences | [
"Big-O Notation",
"Little-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Little-O Notation",
"Definition:Big-O Notation"
] | [
"Category:Big-O Notation",
"Category:Little-O Notation"
] |
proofwiki-13380 | Equivalence of Definitions of Change of Basis Matrix | Let $R$ be a ring with unity.
Let $G$ be a finite-dimensional unitary $R$-module.
Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be ordered bases of $G$.
{{TFAE|def = Change of Basis Matrix}} | It will be shown that the two matrices defined are equal column-wise.
Let $\ds b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$ for $i$ ranging from $1$ to $n$, where $c_{i j}$'s are scalars.
The uniqueness of the above expression is justified by Expression of Vector as Linear Combination from Basis is Unique.
Then by definit... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $G$ be a [[Definition:Dimension (Linear Algebra)|finite-dimensional]] [[Definition:Unitary Module|unitary $R$-module]].
Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be [[Definition:Ordered Basis|ordered bases]] of $G$.
{{TFAE|def = Change of Ba... | It will be shown that the two [[Definition:Matrix|matrices]] defined are equal [[Definition:Column of Matrix|column-wise]].
Let $\ds b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$ for $i$ ranging from $1$ to $n$, where $c_{i j}$'s are [[Definition:Scalar (Module)|scalars]].
The [[Definition:Unique|uniqueness]] of the abo... | Equivalence of Definitions of Change of Basis Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Change_of_Basis_Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Change_of_Basis_Matrix | [
"Linear Algebra",
"Matrix Theory"
] | [
"Definition:Ring with Unity",
"Definition:Dimension (Linear Algebra)",
"Definition:Unitary Module over Ring",
"Definition:Ordered Basis"
] | [
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Scalar/Module",
"Definition:Unique",
"Expression of Vector as Linear Combination from Basis is Unique",
"Definition:Coordinate Vector",
"Definition:Matrix/Column",
"Definition:Change of Basis Matrix/Definition 1",
"Definition:Relative Matr... |
proofwiki-13381 | Substitution in Big-O Estimate/General Result | Let $X$ and $Y$ be topological spaces.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.
Let $x_0 \in X$ and $y_0 \in Y$.
Let $f: X \to Y$ be a function with $\map f {x_0} = y_0$ that is continuous at $x_0$.
Let $g, h: Y \to V$ be functions.
Suppose $\map g y = \map O {\map h y}$ as $y \... | Because $g = \map O h$, there exists a neighborhood $V$ of $y_0$ and a real number $c$ such that:
:$\norm {\map g x} \le c \cdot \norm {\map h x}$ for all $y \in V$.
By definition of continuity, there exists a neighborhood $U$ of $x_0$ with $\map f U \subset V$.
For $x \in U$, we have:
:$\norm {\map g {\map f x} } \le ... | Let $X$ and $Y$ be [[Definition:Topological Space|topological spaces]].
Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$.
Let $x_0 \in X$ and $y_0 \in Y$.
Let $f: X \to Y$ be a function with $\map f {x_0} = y_0$ th... | Because $g = \map O h$, there exists a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] $V$ of $y_0$ and a [[Definition:Real Number|real number]] $c$ such that:
:$\norm {\map g x} \le c \cdot \norm {\map h x}$ for all $y \in V$.
By definition of [[Definition:Continuous Mapping at Point (Topology)... | Substitution in Big-O Estimate/General Result | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/General_Result | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/General_Result | [
"Big-O Notation"
] | [
"Definition:Topological Space",
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Big-O Notation"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Real Number",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Category:Big-O Notation"
] |
proofwiki-13382 | Substitution in Big-O Estimate/Sequences | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
Let $a_n = \map \OO {b_n}$ where $\OO$ denotes big-O notation.
Let $\sequence {n_k}$ be a diverging sequence of natural numbers.
Then $a_{n_k} = \map \OO {b_{n_k} }$. | Because $a_n = \map \OO {b_n}$, there exists $M \ge 0$ and $n_0 \in \N$ such that $\size {a_n} \le M \cdot \size {b_n}$ for $n \ge n_0$.
Because $n_k$ diverges, there exists $k_0 \in \N$ such that $n_k \ge n_0$ for $k \ge k_0$.
Then $\size {a_{n_k} } \le M \cdot \size {b_{n_k} }$ for $k \ge k_0$.
Thus:
:$a_{n_k} = \map... | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
Let $a_n = \map \OO {b_n}$ where $\OO$ denotes [[Definition:Big-O Notation|big-O notation]].
Let $\sequence {n_k}$ be a [[Definition:Divergent Sequence|... | Because $a_n = \map \OO {b_n}$, there exists $M \ge 0$ and $n_0 \in \N$ such that $\size {a_n} \le M \cdot \size {b_n}$ for $n \ge n_0$.
Because $n_k$ [[Definition:Divergent Sequence|diverges]], there exists $k_0 \in \N$ such that $n_k \ge n_0$ for $k \ge k_0$.
Then $\size {a_{n_k} } \le M \cdot \size {b_{n_k} }$ for... | Substitution in Big-O Estimate/Sequences | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/Sequences | https://proofwiki.org/wiki/Substitution_in_Big-O_Estimate/Sequences | [
"Big-O Notation"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Big-O Notation",
"Definition:Divergent Sequence",
"Definition:Natural Numbers"
] | [
"Definition:Divergent Sequence",
"Category:Big-O Notation"
] |
proofwiki-13383 | First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values | The first sequence of $3$ consecutive positive integers whose Euler $\phi$ values are strictly decreasing is:
:$\map \phi {523} > \map \phi {524} > \map \phi {525}$ | {{begin-eqn}}
{{eqn | l = \map \phi {523}
| r = 522
| c = Euler Phi Function of Prime
}}
{{eqn | l = \map \phi {524}
| r = 260
| c = {{EulerPhiLink|524}}
}}
{{eqn | l = \map \phi {525}
| r = 240
| c = {{EulerPhiLink|525}}
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that this ... | The first [[Definition:Integer Sequence|sequence]] of $3$ consecutive [[Definition:Positive Integer|positive integers]] whose [[Definition:Euler Phi Function|Euler $\phi$ values]] are [[Definition:Strictly Decreasing Sequence|strictly decreasing]] is:
:$\map \phi {523} > \map \phi {524} > \map \phi {525}$ | {{begin-eqn}}
{{eqn | l = \map \phi {523}
| r = 522
| c = [[Euler Phi Function of Prime]]
}}
{{eqn | l = \map \phi {524}
| r = 260
| c = {{EulerPhiLink|524}}
}}
{{eqn | l = \map \phi {525}
| r = 240
| c = {{EulerPhiLink|525}}
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that ... | First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values | https://proofwiki.org/wiki/First_Sequence_of_Three_Consecutive_Strictly_Decreasing_Euler_Phi_Values | https://proofwiki.org/wiki/First_Sequence_of_Three_Consecutive_Strictly_Decreasing_Euler_Phi_Values | [
"Euler Phi Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Euler Phi Function",
"Definition:Strictly Decreasing/Sequence"
] | [
"Euler Phi Function of Prime"
] |
proofwiki-13384 | Bases of Free Module have Equal Cardinality | Let $R$ be a commutative ring with unity.
Let $M$ be a free $R$-module.
Let $B$ and $C$ be bases of $M$.
Then $B$ and $C$ are equivalent.
That is, they have the same cardinality. | By definition, a basis is a generator.
By Basis of Free Module is No Greater than Generator, there exist:
:an injection $f : B \to C$
:an injection $g : C \to B$
By the Cantor-Bernstein-Schröder Theorem, $B$ and $C$ are equivalent.
{{qed}} | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]].
Let $B$ and $C$ be [[Definition:Basis of Module|bases]] of $M$.
Then $B$ and $C$ are [[Definition:Set Equivalence|equivalent]].
That is, they have the same [[Defin... | By definition, a [[Definition:Basis of Module|basis]] is a [[Definition:Generator of Module|generator]].
By [[Basis of Free Module is No Greater than Generator]], there exist:
:an [[Definition:Injection|injection]] $f : B \to C$
:an [[Definition:Injection|injection]] $g : C \to B$
By the [[Cantor-Bernstein-Schröder T... | Bases of Free Module have Equal Cardinality | https://proofwiki.org/wiki/Bases_of_Free_Module_have_Equal_Cardinality | https://proofwiki.org/wiki/Bases_of_Free_Module_have_Equal_Cardinality | [
"Free Modules",
"Bases of Modules"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Free Module over Ring",
"Definition:Basis of Module",
"Definition:Set Equivalence",
"Definition:Cardinality"
] | [
"Definition:Basis of Module",
"Definition:Generator of Module",
"Basis of Free Module is No Greater than Generator",
"Definition:Injection",
"Definition:Injection",
"Cantor-Bernstein-Schröder Theorem",
"Definition:Set Equivalence"
] |
proofwiki-13385 | Sequences of Three Consecutive Strictly Increasing Euler Phi Values | The following sequences of $3$ consecutive positive integers have the property that their Euler $\phi$ values are strictly increasing:
:$\tuple {105, 106, 107}, \tuple {165, 166, 167}, \tuple {315, 316, 317}, \tuple {525, 526, 527}, \dots$
{{expand|Related sequences: A161962 (superset), A161963}} | {{begin-eqn}}
{{eqn | l = \map \phi {525}
| r = 240
| c = {{EulerPhiLink|525}}
}}
{{eqn | l = \map \phi {526}
| r = 262
| c = {{EulerPhiLink|526}}
}}
{{eqn | l = \map \phi {527}
| r = 480
| c = {{EulerPhiLink|527}}
}}
{{end-eqn}}
{{finish}} | The following [[Definition:Integer Sequence|sequences]] of $3$ consecutive [[Definition:Positive Integer|positive integers]] have the property that their [[Definition:Euler Phi Function|Euler $\phi$ values]] are [[Definition:Strictly Increasing Sequence|strictly increasing]]:
:$\tuple {105, 106, 107}, \tuple {165, 166,... | {{begin-eqn}}
{{eqn | l = \map \phi {525}
| r = 240
| c = {{EulerPhiLink|525}}
}}
{{eqn | l = \map \phi {526}
| r = 262
| c = {{EulerPhiLink|526}}
}}
{{eqn | l = \map \phi {527}
| r = 480
| c = {{EulerPhiLink|527}}
}}
{{end-eqn}}
{{finish}} | Sequences of Three Consecutive Strictly Increasing Euler Phi Values | https://proofwiki.org/wiki/Sequences_of_Three_Consecutive_Strictly_Increasing_Euler_Phi_Values | https://proofwiki.org/wiki/Sequences_of_Three_Consecutive_Strictly_Increasing_Euler_Phi_Values | [
"Euler Phi Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Euler Phi Function",
"Definition:Strictly Increasing/Sequence"
] | [] |
proofwiki-13386 | Basis of Vector Space Injects into Generator | Let $K$ be a division ring.
Let $V$ be a vector space over $K$.
Let $B$ be a basis of $V$.
Let $G$ be a generator of $V$.
Then there exists an injection from $B$ to $G$. | By Vector Space has Basis between Linearly Independent Set and Spanning Set, there exists a basis $C \subset G$.
By Bases of Vector Space have Equal Cardinality, there exists a bijection between $B$ and $C$.
By Composite of Injections is Injection, composing this bijection with the inclusion of $C$ in $G$, we obtain an... | Let $K$ be a [[Definition:Division Ring|division ring]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $B$ be a [[Definition:Basis of Vector Space|basis]] of $V$.
Let $G$ be a [[Definition:Generator of Vector Space|generator]] of $V$.
Then there exists an [[Definition:Injection|injection]] fr... | By [[Vector Space has Basis between Linearly Independent Set and Spanning Set]], there exists a [[Definition:Basis of Vector Space|basis]] $C \subset G$.
By [[Bases of Vector Space have Equal Cardinality]], there exists a [[Definition:Bijection|bijection]] between $B$ and $C$.
By [[Composite of Injections is Injectio... | Basis of Vector Space Injects into Generator | https://proofwiki.org/wiki/Basis_of_Vector_Space_Injects_into_Generator | https://proofwiki.org/wiki/Basis_of_Vector_Space_Injects_into_Generator | [
"Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Basis of Vector Space",
"Definition:Generator of Vector Space",
"Definition:Injection"
] | [
"Vector Space has Basis between Linearly Independent Set and Spanning Set",
"Definition:Basis of Vector Space",
"Bases of Vector Space have Equal Cardinality",
"Definition:Bijection",
"Composite of Injections is Injection",
"Definition:Bijection",
"Definition:Inclusion Mapping",
"Definition:Injection"... |
proofwiki-13387 | Increasing Union of Ideals is Ideal/Sequence | Let $R$ be a ring.
Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be ideals of $R$.
Then the increasing union $S$:
:$\ds S = \bigcup_{i \mathop \in \N} S_i$
is an ideal of $R$. | Let $\ds S = \bigcup_{i \mathop \in \N} S_i$.
From Increasing Union of Subrings is Subring, we have that $S$ is a subring of $R$.
Now we need to show that it is an ideal of $R$.
Let $a \in S$.
Then $\exists i \in \N: a \in S_i$.
Let $b \in R$.
Then $a b \in S_i$ and $b a \in S_i$, as $S_i$ is an ideal of $R$.
Thus $a b... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be [[Definition:Ideal of Ring|ideals]] of $R$.
Then the [[Definition:Increasing Union|increasing union]] $S$:
:$\ds S = \bigcup_{i \mathop \in \N} S_i$
is an [[Definition:Id... | Let $\ds S = \bigcup_{i \mathop \in \N} S_i$.
From [[Increasing Union of Subrings is Subring]], we have that $S$ is a [[Definition:Subring|subring]] of $R$.
Now we need to show that it is an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $a \in S$.
Then $\exists i \in \N: a \in S_i$.
Let $b \in R$.
Then $a b \in... | Increasing Union of Ideals is Ideal/Sequence | https://proofwiki.org/wiki/Increasing_Union_of_Ideals_is_Ideal/Sequence | https://proofwiki.org/wiki/Increasing_Union_of_Ideals_is_Ideal/Sequence | [
"Set Union",
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Increasing Union",
"Definition:Ideal of Ring"
] | [
"Increasing Union of Subrings is Subring",
"Definition:Subring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring"
] |
proofwiki-13388 | Increasing Union of Ideals is Ideal/Chain | Let $R$ be a ring.
Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion.
Let $\set {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of ideals in $P$.
Let $\ds I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union.
Then $I$ is an ideal of $R$. | === Property 1: $0 \in I$ ===
Since $\set {I_\alpha}_{\alpha \mathop \in A}$ is non-empty chain, it must contain some ideal $I_\beta$
Since $I_\beta$ is an ideal, $0 \in I_\beta$.
Thus $0 \in I$. | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {P, \subseteq}$ be the [[Definition:Ordered Set|ordered set]] consisting of all [[Definition:Ideal of Ring|ideals]] of $R$, ordered by [[Definition:Subset|inclusion]].
Let $\set {I_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Non-Empty Set|no... | === Property 1: $0 \in I$ ===
Since $\set {I_\alpha}_{\alpha \mathop \in A}$ is [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain of Sets|chain]], it must contain some [[Definition:Ideal of Ring|ideal]] $I_\beta$
Since $I_\beta$ is an [[Definition:Ideal of Ring|ideal]], $0 \in I_\beta$.
Thus $0 \in I$. | Increasing Union of Ideals is Ideal/Chain | https://proofwiki.org/wiki/Increasing_Union_of_Ideals_is_Ideal/Chain | https://proofwiki.org/wiki/Increasing_Union_of_Ideals_is_Ideal/Chain | [
"Ring Theory",
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ordered Set",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Set Union",
"Definition:Ideal of Ring"
] | [
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring"
] |
proofwiki-13389 | Union of Chain of Proper Ideals is Proper Ideal | Let $R$ be a ring with unity.
Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion.
Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of proper ideals in $P$.
Let $\ds I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union.
Then $I$ is a proper ... | By Union of Chain of Ideals is Ideal, $I$ is an ideal.
It remains to show that $I \subsetneq R$.
The ideals $I_\alpha$ are all proper, so none of them contain the unity.
Thus $I$ does not contain $1$, which means $I \subsetneq R$.
{{qed}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $\struct {P, \subseteq}$ be the [[Definition:Ordered Set|ordered set]] consisting of all [[Definition:Ideal of Ring|ideals]] of $R$, ordered by [[Definition:Subset|inclusion]].
Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Non-Empt... | By [[Union of Chain of Ideals is Ideal]], $I$ is an [[Definition:Ideal of Ring|ideal]].
It remains to show that $I \subsetneq R$.
The [[Definition:Ideal of Ring|ideals]] $I_\alpha$ are all [[Definition:Proper Ideal of Ring|proper]], so none of them contain the [[Definition:Unity of Ring|unity]].
Thus $I$ does not co... | Union of Chain of Proper Ideals is Proper Ideal | https://proofwiki.org/wiki/Union_of_Chain_of_Proper_Ideals_is_Proper_Ideal | https://proofwiki.org/wiki/Union_of_Chain_of_Proper_Ideals_is_Proper_Ideal | [
"Ring Theory",
"Ideal Theory"
] | [
"Definition:Ring with Unity",
"Definition:Ordered Set",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Nested Sequence",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Set Union",
"Definition:Ideal of Ring/Proper Ideal"
] | [
"Increasing Union of Ideals is Ideal/Chain",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unity (Abstract Algebra)/Ring"
] |
proofwiki-13390 | Dimension of Free Vector Space on Set | Let $k$ be a division ring.
Let $X$ be a set.
Let $k^{\paren X}$ be the free vector space on $X$.
The vector space $k^{\paren X}$ has dimension the cardinality of $X$. | Follows from:
:Canonical Basis of Free Module on Set is Basis
:Cardinality of Canonical Basis of Free Module on Set
{{qed}}
Category:Vector Spaces
tfppnzuta22hqh19b9ep3xu1k0mgyru | Let $k$ be a [[Definition:Division Ring|division ring]].
Let $X$ be a [[Definition:Set|set]].
Let $k^{\paren X}$ be the [[Definition:Free Vector Space on Set|free vector space]] on $X$.
The [[Definition:Vector Space|vector space]] $k^{\paren X}$ has [[Definition:Dimension of Vector Space|dimension]] the [[Definitio... | Follows from:
:[[Canonical Basis of Free Module on Set is Basis]]
:[[Cardinality of Canonical Basis of Free Module on Set]]
{{qed}}
[[Category:Vector Spaces]]
tfppnzuta22hqh19b9ep3xu1k0mgyru | Dimension of Free Vector Space on Set | https://proofwiki.org/wiki/Dimension_of_Free_Vector_Space_on_Set | https://proofwiki.org/wiki/Dimension_of_Free_Vector_Space_on_Set | [
"Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Set",
"Definition:Free Module on Set",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Cardinality"
] | [
"Canonical Basis of Free Module on Set is Basis",
"Cardinality of Canonical Basis of Free Module on Set",
"Category:Vector Spaces"
] |
proofwiki-13391 | Numbers not Expressible as Sum of Fewer than 19 Fourth Powers | The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers:
:$79, 159, 239, 319, 399, 479, 559$
{{OEIS|A046050}} | On a case-by-case basis:
From Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers:
{{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}}
From 159 is not Expressible as Sum of Fewer than 19 Fourth Powers:
{{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
From 239 is... | The following [[Definition:Positive Integer|positive integer]] are the only ones which cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]]:
:$79, 159, 239, 319, 399, 479, 559$
{{OEIS|A046050}} | On a case-by-case basis:
From [[Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers]]:
{{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}}
From [[159 is not Expressible as Sum of Fewer than 19 Fourth Powers]]:
{{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
... | Numbers not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [
"Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers",
"159 is not Expressible as Sum of Fewer than 19 Fourth Powers",
"239 is not Expressible as Sum of Fewer than 19 Fourth Powers",
"319 is not Expressible as Sum of Fewer than 19 Fourth Powers",
"399 is not Expressible as Sum of Fewer tha... |
proofwiki-13392 | 239 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$ | First note that $4^4 = 256 > 239$.
Then note that $3 \times 3^4 = 243 > 239$.
Hence any expression of $239$ as fourth powers uses no $n^4$ for $n \ge 4$, and uses not more than $2$ instances $3^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances $1^4$ does.
Now we have:
{{begin-eqn}}
{{eqn |... | :$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$ | First note that $4^4 = 256 > 239$.
Then note that $3 \times 3^4 = 243 > 239$.
Hence any expression of $239$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 4$, and uses not more than $2$ instances $3^4$.
For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $1... | 239 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/239_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/239_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"239"
] | [] | [
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power"
] |
proofwiki-13393 | 479 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$
or:
:$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$ | First note that $5^4 = 625 > 479$.
Then note that $2 \times 4^4 = 512 > 479$.
Hence any expression of $479$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
{{begin-eqn}}
{{... | :$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$
or:
:$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$ | First note that $5^4 = 625 > 479$.
Then note that $2 \times 4^4 = 512 > 479$.
Hence any expression of $479$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than ... | 479 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/479_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/479_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"479"
] | [] | [
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power"
] |
proofwiki-13394 | 559 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$
or:
:$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$ | First note that $5^4 = 625 > 559$.
Then note that $3 \times 4^4 = 768 > 559$.
Hence any expression of $559$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $2$ instances of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
{{begin-eqn}}
{... | :$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$
or:
:$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$ | First note that $5^4 = 625 > 559$.
Then note that $3 \times 4^4 = 768 > 559$.
Hence any expression of $559$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $2$ instances of $4^4$.
For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than... | 559 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/559_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/559_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"559"
] | [] | [
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power"
] |
proofwiki-13395 | Free Module on Set is Free | Let $R$ be a ring with unity.
Let $I$ be a set.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Then $R^{\paren I}$ is a free $R$-module. | From Canonical Basis of Free Module on Set is Basis, $R^{\paren I}$ has a basis.
{{qed}}
Category:Free Modules
jpeodkx099o9yv0mnusg2golguk0j9d | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $I$ be a [[Definition:Set|set]].
Let $R^{\paren I}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
Then $R^{\paren I}$ is a [[Definition:Free Module over Ring|free $R$-module]]. | From [[Canonical Basis of Free Module on Set is Basis]], $R^{\paren I}$ has a [[Definition:Basis of Module|basis]].
{{qed}}
[[Category:Free Modules]]
jpeodkx099o9yv0mnusg2golguk0j9d | Free Module on Set is Free | https://proofwiki.org/wiki/Free_Module_on_Set_is_Free | https://proofwiki.org/wiki/Free_Module_on_Set_is_Free | [
"Free Modules"
] | [
"Definition:Ring with Unity",
"Definition:Set",
"Definition:Free Module on Set",
"Definition:Free Module over Ring"
] | [
"Canonical Basis of Free Module on Set is Basis",
"Definition:Basis of Module",
"Category:Free Modules"
] |
proofwiki-13396 | Canonical Basis of Free Module on Set is Basis | Let $R$ be a ring with unity.
Let $I$ be a set.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Let $B$ be its canonical basis.
Then $B$ is a basis of $R^{\paren I}$. | {{refactor|I'm not sure it's completely necessary just to repeat everything in the exposition}}
Let $R$ be a ring with unity.
Let $I$ be a set.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Let $B$ be its canonical basis.
Recall the definition of canonical basis:
{{:Definition:Canonical Basis of Free Module on Set}... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $I$ be a [[Definition:Set|set]].
Let $R^{\paren I}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
Let $B$ be its [[Definition:Canonical Basis of Free Module on Set|canonical basis]].
Then $B$ is a [[Definition:Basis of Module|basis... | {{refactor|I'm not sure it's completely necessary just to repeat everything in the exposition}}
Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $I$ be a [[Definition:Set|set]].
Let $R^{\paren I}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
Let $B$ be its [[Definition:Canonical ... | Canonical Basis of Free Module on Set is Basis | https://proofwiki.org/wiki/Canonical_Basis_of_Free_Module_on_Set_is_Basis | https://proofwiki.org/wiki/Canonical_Basis_of_Free_Module_on_Set_is_Basis | [
"Free Modules"
] | [
"Definition:Ring with Unity",
"Definition:Set",
"Definition:Free Module on Set",
"Definition:Canonical Basis of Free Module on Set",
"Definition:Basis of Module"
] | [
"Definition:Ring with Unity",
"Definition:Set",
"Definition:Free Module on Set",
"Definition:Canonical Basis of Free Module on Set",
"Definition:Canonical Basis of Free Module on Set",
"Definition:Basis of Module",
"Category:Free Modules"
] |
proofwiki-13397 | Korselt's Theorem | Let $n \ge 2$ be an integer.
Then $n$ is a '''Carmichael number''' {{iff}}:
: $(1): \quad n$ is odd
and the following conditions hold for every prime factor $p$ of $n$:
: $(2): \quad p^2 \nmid n$
: $(3): \quad \paren {p - 1} \divides \paren {n - 1}$
where:
:$\divides$ denotes divisibility
:$\nmid$ denotes non-divisibil... | === Sufficient Condition ===
Let $n$ be a Carmichael number:
:$(4): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$
where $\perp$ denotes coprimality.
Suppose $n$ is even.
Set $a = -1$ in $(4)$.
Then $\paren {-1}^n = 1$ and so:
:$1 \equiv -1 \pmod n$
resulting in $n = 2$.
But as $2$ is not a Carmichael number,... | Let $n \ge 2$ be an [[Definition:Integer|integer]].
Then $n$ is a '''[[Definition:Carmichael Number|Carmichael number]]''' {{iff}}:
: $(1): \quad n$ is [[Definition:Odd Integer|odd]]
and the following conditions hold for every [[Definition:Prime Factor|prime factor]] $p$ of $n$:
: $(2): \quad p^2 \nmid n$
: $(3): \qua... | === Sufficient Condition ===
Let $n$ be a [[Definition:Carmichael Number|Carmichael number]]:
:$(4): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Suppose $n$ is [[Definition:Even Integer|even]].
Set $a = -1$ in $(4)$.
Then $\paren {-1... | Korselt's Theorem | https://proofwiki.org/wiki/Korselt's_Theorem | https://proofwiki.org/wiki/Korselt's_Theorem | [
"Carmichael Numbers"
] | [
"Definition:Integer",
"Definition:Carmichael Number",
"Definition:Odd Integer",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Carmichael Number",
"Definition:Coprime/Integers",
"Definition:Even Integer",
"Definition:Carmichael Number",
"Definition:Carmichael Number",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Fundamental Theorem of Arithmetic",
"Definition:Distinct",
"Definition:Odd Prime",
"Condi... |
proofwiki-13398 | Carmichael Number has 3 Odd Prime Factors | Let $n$ be a Carmichael number.
Then $n$ has at least $3$ distinct odd prime factors. | By Korselt's Theorem, $n$ is odd.
Therefore $n$ has at least $1$ odd prime factor.
By Korselt's Theorem, for each prime factor of $n$:
:$p^2 \nmid n$
:$\paren {p - 1} \divides \paren {n - 1}$
Suppose $n = p^k$ for some odd prime $p$.
By Korselt's Theorem, $k = 1$.
However by definition of a Carmichael Number, $n$ canno... | Let $n$ be a [[Definition:Carmichael Number|Carmichael number]].
Then $n$ has at least $3$ [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factors]]. | By [[Korselt's Theorem]], $n$ is [[Definition:Odd Integer|odd]].
Therefore $n$ has at least $1$ [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factor]].
By [[Korselt's Theorem]], for each [[Definition:Prime Factor|prime factor]] of $n$:
:$p^2 \nmid n$
:$\paren {p - 1} \divides \paren {n - 1}$
Suppose... | Carmichael Number has 3 Odd Prime Factors | https://proofwiki.org/wiki/Carmichael_Number_has_3_Odd_Prime_Factors | https://proofwiki.org/wiki/Carmichael_Number_has_3_Odd_Prime_Factors | [
"Carmichael Numbers"
] | [
"Definition:Carmichael Number",
"Definition:Distinct",
"Definition:Odd Prime",
"Definition:Prime Factor"
] | [
"Korselt's Theorem",
"Definition:Odd Integer",
"Definition:Odd Prime",
"Definition:Prime Factor",
"Korselt's Theorem",
"Definition:Prime Factor",
"Definition:Odd Prime",
"Definition:Prime Number",
"Korselt's Theorem",
"Definition:Carmichael Number",
"Definition:Prime Number",
"Definition:Disti... |
proofwiki-13399 | Intersection of Submodules is Submodule/General Result | Let $S$ be a set of submodules of $M$.
Then the intersection $\bigcap S$ is a submodule of $M$. | From Intersection of Subgroups is Subgroup:General Result, it follows that $\bigcap S$ is a subgroup of $M$.
As a subgroup is closed for its operation, it follows that for all $x, y \in \bigcap S$, we have $x + y, y + x \in \bigcap S$.
As $M$ is an $R$-module, and the addition $+$ on $\bigcap S$ is the restriction of t... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Submodule|submodules]] of $M$.
Then the [[Definition:Intersection of Family|intersection]] $\bigcap S$ is a [[Definition:Submodule|submodule]] of $M$. | From [[Intersection of Subgroups is Subgroup/General Result|Intersection of Subgroups is Subgroup:General Result]], it follows that $\bigcap S$ is a [[Definition:Subgroup|subgroup]] of $M$.
As a [[Definition:Subgroup|subgroup]] is [[Definition:Closed Algebraic Structure|closed for its operation]], it follows that for ... | Intersection of Submodules is Submodule/General Result | https://proofwiki.org/wiki/Intersection_of_Submodules_is_Submodule/General_Result | https://proofwiki.org/wiki/Intersection_of_Submodules_is_Submodule/General_Result | [
"Module Theory"
] | [
"Definition:Set",
"Definition:Submodule",
"Definition:Set Intersection/Family of Sets",
"Definition:Submodule"
] | [
"Intersection of Subgroups is Subgroup/General Result",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Module over Ring",
"Definition:Additive Notation",
"Definition:Restriction/Operation",
"Definition:Additive Notation",
"Defin... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.