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-3100 | Projection onto Ideal of External Direct Sum of Rings | Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$ be rings.
Let $\ds \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$ be their direct product.
For each $k \in \closedint 1 n$, let:
:${R_k}' = \set {\tuple {x_1, \ldots, x_n} \in R: \forall j \... | From Ideal of External Direct Sum of Rings we already have that ${R_k}'$ is an ideal of $R$.
The result follows by application of Projection is Epimorphism.
{{qed}} | Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$ be [[Definition:Ring (Abstract Algebra)|rings]].
Let $\ds \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$ be their [[Definition:Ring Direct Product|direct product]].
For each $k \in \close... | From [[Ideal of External Direct Sum of Rings]] we already have that ${R_k}'$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
The result follows by application of [[Projection is Epimorphism]].
{{qed}} | Projection onto Ideal of External Direct Sum of Rings | https://proofwiki.org/wiki/Projection_onto_Ideal_of_External_Direct_Sum_of_Rings | https://proofwiki.org/wiki/Projection_onto_Ideal_of_External_Direct_Sum_of_Rings | [
"Ideal Theory",
"Projections"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Direct Product",
"Definition:Projection (Mapping Theory)",
"Definition:Ring Epimorphism"
] | [
"Ideal of External Direct Sum of Rings",
"Definition:Ideal of Ring",
"Projection is Epimorphism"
] |
proofwiki-3101 | Conditions for Internal Ring Direct Sum | Let $\struct {R, +, \circ}$ be a ring.
Let $\sequence {\struct {S_k, +, \circ} }$ be a sequence of subrings of $R$.
Then $R$ is the ring direct sum of $\sequence {S_k}_{1 \mathop \le k \mathop \le n}$ {{iff}}:
:$(1): \quad R = S_1 + S_2 + \cdots + S_n$
:$(2): \quad \sequence {\struct {S_k, +} }_{1 \mathop \le k \mathop... | Let $S$ be the cartesian product of $\sequence {\struct {S_k, +} }$
First note that $\phi$ is a group homomorphism from $\struct {S, +}$ to $\struct {R, +}$, as:
:$\ds \sum_{j \mathop = 1}^n \paren {x_j + y_j} = \sum_{j \mathop = 1}^n x_j + \sum_{j \mathop = 1}^n y_j$ from Associativity on Indexing Set.
So $\phi$ is a ... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\sequence {\struct {S_k, +, \circ} }$ be a [[Definition:Sequence|sequence]] of [[Definition:Subring|subrings]] of $R$.
Then $R$ is the [[Definition:Ring Direct Sum|ring direct sum]] of $\sequence {S_k}_{1 \mathop \le k \mathop \le n}$... | Let $S$ be the [[Definition:Cartesian Product|cartesian product]] of $\sequence {\struct {S_k, +} }$
First note that $\phi$ is a [[Definition:Group Homomorphism|group homomorphism]] from $\struct {S, +}$ to $\struct {R, +}$, as:
:$\ds \sum_{j \mathop = 1}^n \paren {x_j + y_j} = \sum_{j \mathop = 1}^n x_j + \sum_{j \ma... | Conditions for Internal Ring Direct Sum | https://proofwiki.org/wiki/Conditions_for_Internal_Ring_Direct_Sum | https://proofwiki.org/wiki/Conditions_for_Internal_Ring_Direct_Sum | [
"Subrings",
"Ideal Theory",
"Direct Sums of Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Sequence",
"Definition:Subring",
"Definition:Internal Direct Sum of Rings",
"Definition:Sequence",
"Definition:Independent Subgroups",
"Definition:Ideal of Ring"
] | [
"Definition:Cartesian Product",
"Definition:Group Homomorphism",
"Associativity on Indexing Set",
"Definition:Ring Homomorphism",
"Definition:Cartesian Product",
"Definition:Mapping",
"Definition:Surjection",
"Internal Direct Product Generated by Subgroups",
"Definition:Isomorphism (Abstract Algebra... |
proofwiki-3102 | Structure Induced by Ring Operations is Ring | Let $\struct {R, +, \circ}$ be a ring.
Let $S$ be a set.
Then $\struct {R^S, +', \circ'}$ is a ring, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$. | As $R$ is a ring, both $+$ and $\circ$ are closed on $R$ by definition.
From Closure of Pointwise Operation on Algebraic Structure, it follows that both $+'$ and $\circ'$ are closed on $R^S$:
:$\forall f, g \in R^S: f +' g \in R^S$
:$\forall f, g \in R^S: f \circ' g \in R^S$
By Structure Induced by Abelian Group Operat... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $S$ be a [[Definition:Set|set]].
Then $\struct {R^S, +', \circ'}$ is a [[Definition:Ring (Abstract Algebra)|ring]], where $+'$ and $\circ'$ are the [[Definition:Pointwise Operation|pointwise operations induced]] on $R^S$ by $+$ and $\c... | As $R$ is a [[Definition:Ring (Abstract Algebra)|ring]], both $+$ and $\circ$ are [[Definition:Closed Operation|closed]] on $R$ by definition.
From [[Closure of Pointwise Operation on Algebraic Structure]], it follows that both $+'$ and $\circ'$ are [[Definition:Closed Operation|closed]] on $R^S$:
:$\forall f, g \in ... | Structure Induced by Ring Operations is Ring | https://proofwiki.org/wiki/Structure_Induced_by_Ring_Operations_is_Ring | https://proofwiki.org/wiki/Structure_Induced_by_Ring_Operations_is_Ring | [
"Rings of Mappings",
"Examples of Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Set",
"Definition:Ring (Abstract Algebra)",
"Definition:Pointwise Operation"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Closure of Pointwise Operation on Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Structure Induced by Abelian Group Operation is Abelian Group",
"Definition:Abelian Grou... |
proofwiki-3103 | Division Subring Test | Let $\struct {K, +, \circ}$ be a division ring, and let $L$ be a subset of $K$.
Then $\struct {L, +, \circ}$ is a division subring of $\struct {K, +, \circ}$ {{iff}} these all hold:
:$(1) \quad L^* \ne \O$
:$(2) \quad \forall x, y \in L: x + \paren {-y} \in L$
:$(3) \quad \forall x, y \in L: x \circ y \in L$
:$(4) \qua... | === Necessary Condition ===
Suppose $\struct {L, +, \circ}$ is a division subring of $\struct {K, +, \circ}$.
The conditions $(1)$ to $(3)$ hold by virtue of the Subring Test.
Then $(4)$ also holds by the definition of a division ring:
:$\forall x \in L^*: \exists ! x^{-1} \in L^*: x^{-1} \circ x = x \circ x^{-1} = 1_L... | Let $\struct {K, +, \circ}$ be a [[Definition:Division Ring|division ring]], and let $L$ be a [[Definition:Subset|subset]] of $K$.
Then $\struct {L, +, \circ}$ is a [[Definition:Division Subring|division subring]] of $\struct {K, +, \circ}$ {{iff}} these all hold:
:$(1) \quad L^* \ne \O$
:$(2) \quad \forall x, y \i... | === Necessary Condition ===
Suppose $\struct {L, +, \circ}$ is a [[Definition:Division Subring|division subring]] of $\struct {K, +, \circ}$.
The conditions $(1)$ to $(3)$ hold by virtue of the [[Subring Test]].
Then $(4)$ also holds by the definition of a [[Definition:Division Ring|division ring]]:
:$\forall x \in ... | Division Subring Test | https://proofwiki.org/wiki/Division_Subring_Test | https://proofwiki.org/wiki/Division_Subring_Test | [
"Subrings"
] | [
"Definition:Division Ring",
"Definition:Subset",
"Definition:Division Subring"
] | [
"Definition:Division Subring",
"Subring Test",
"Definition:Division Ring",
"Subring Test",
"Definition:Division Ring"
] |
proofwiki-3104 | Set of Division Subrings forms Complete Lattice | Let $\struct {D, +, \circ}$ be a division ring.
Let $\mathbb K$ be the set of all division subrings of $K$.
Then $\struct {\mathbb K, \subseteq}$ is a complete lattice. | Let $\O \subset \mathbb S \subseteq \mathbb K$.
By Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings:
:$\bigcap \mathbb S$ is the largest division subring of $K$ contained in each of the elements of $\mathbb S$.
By Intersection of Division Subrings Containing Subset is Sma... | Let $\struct {D, +, \circ}$ be a [[Definition:Division Ring|division ring]].
Let $\mathbb K$ be the [[Definition:Set|set]] of all [[Definition:Division Subring|division subrings]] of $K$.
Then $\struct {\mathbb K, \subseteq}$ is a [[Definition:Complete Lattice|complete lattice]]. | Let $\O \subset \mathbb S \subseteq \mathbb K$.
By [[Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings]]:
:$\bigcap \mathbb S$ is the largest [[Definition:Division Subring|division subring]] of $K$ contained in each of the elements of $\mathbb S$.
By [[Intersection of D... | Set of Division Subrings forms Complete Lattice | https://proofwiki.org/wiki/Set_of_Division_Subrings_forms_Complete_Lattice | https://proofwiki.org/wiki/Set_of_Division_Subrings_forms_Complete_Lattice | [
"Complete Lattices",
"Division Subrings"
] | [
"Definition:Division Ring",
"Definition:Set",
"Definition:Division Subring",
"Definition:Complete Lattice"
] | [
"Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings",
"Definition:Division Subring",
"Intersection of Division Subrings Containing Subset is Smallest",
"Definition:Set Intersection",
"Definition:Division Subring",
"Definition:Division Subring",
"Definition:L... |
proofwiki-3105 | Set of Subfields forms Complete Lattice | Let $\struct {F, +, \circ}$ be a field.
Let $\mathbb F$ be the set of all subfields of $F$.
Then $\struct {\mathbb F, \subseteq}$ is a complete lattice. | Let $\O \subset \mathbb S \subseteq \mathbb F$.
By Intersection of Subfields is Largest Subfield Contained in all Subfields:
:$\bigcap \mathbb S$ is the largest subfield of $F$ contained in each of the elements of $\mathbb S$.
By Intersection of Subfields Containing Subset is Smallest:
:The intersection of the set of a... | Let $\struct {F, +, \circ}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\mathbb F$ be the [[Definition:Set|set]] of all [[Definition:Subfield|subfields]] of $F$.
Then $\struct {\mathbb F, \subseteq}$ is a [[Definition:Complete Lattice|complete lattice]]. | Let $\O \subset \mathbb S \subseteq \mathbb F$.
By [[Intersection of Subfields is Largest Subfield Contained in all Subfields]]:
:$\bigcap \mathbb S$ is the largest [[Definition:Subfield|subfield]] of $F$ contained in each of the elements of $\mathbb S$.
By [[Intersection of Subfields Containing Subset is Smallest]]... | Set of Subfields forms Complete Lattice | https://proofwiki.org/wiki/Set_of_Subfields_forms_Complete_Lattice | https://proofwiki.org/wiki/Set_of_Subfields_forms_Complete_Lattice | [
"Complete Lattices",
"Subfields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Set",
"Definition:Subfield",
"Definition:Complete Lattice"
] | [
"Intersection of Subfields is Largest Subfield Contained in all Subfields",
"Definition:Subfield",
"Intersection of Subfields Containing Subset is Smallest",
"Definition:Set Intersection",
"Definition:Subfield",
"Definition:Subfield",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set",
"De... |
proofwiki-3106 | Properties of Prime Subfield | Let $F$ be a field.
Let $K$ be the prime subfield of $F$.
Then $K$ is isomorphic to either:
:$\Q$, the field of rational numbers, or
:$\Z_p$, the Ring of Integers Modulo $p$, where $p$ is prime. | From Field of Characteristic Zero has Unique Prime Subfield, if $\Char F = 0$, then its prime subfield is isomorphic to $\Q$, the field of rational numbers.
From Field of Prime Characteristic has Unique Prime Subfield, if $\Char F = p$, then its prime subfield is isomorphic to $\Z_p$, the Ring of Integers Modulo $p$.
F... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $K$ be the [[Definition:Prime Subfield|prime subfield]] of $F$.
Then $K$ is [[Definition:Field Isomorphism|isomorphic]] to either:
:$\Q$, the [[Definition:Field of Rational Numbers|field of rational numbers]], or
:$\Z_p$, the [[Ring of Integers Modulo... | From [[Field of Characteristic Zero has Unique Prime Subfield]], if $\Char F = 0$, then its [[Definition:Prime Subfield|prime subfield]] is [[Definition:Field Isomorphism|isomorphic]] to $\Q$, the [[Definition:Field of Rational Numbers|field of rational numbers]].
From [[Field of Prime Characteristic has Unique Prime ... | Properties of Prime Subfield | https://proofwiki.org/wiki/Properties_of_Prime_Subfield | https://proofwiki.org/wiki/Properties_of_Prime_Subfield | [
"Subfields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Prime Subfield",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Field of Rational Numbers",
"Ring of Integers Modulo Prime is Field",
"Definition:Prime Number"
] | [
"Field of Characteristic Zero has Unique Prime Subfield",
"Definition:Prime Subfield",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Field of Rational Numbers",
"Field of Prime Characteristic has Unique Prime Subfield",
"Definition:Prime Subfield",
"Definition:Isomorphism (A... |
proofwiki-3107 | Subgroup of Additive Group Modulo m is Ideal of Ring | Let $m \in \Z: m > 1$.
Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$.
Then every subgroup of $\struct {\Z_m, +_m}$ is an ideal of the ring of integers modulo $m$ $\struct {\Z_m, +_m, \times_m}$. | Let $H$ be a subgroup of $\struct {\Z_m, +_m}$
Suppose:
: $(1): \quad h + \ideal m \in H$, where $\ideal m$ is a principal ideal of $\struct {\Z_m, +_m, \times_m}$
and
: $(2): \quad n \in \N_{>0}$.
Then by definition of multiplication on integers and Homomorphism of Powers as applied to integers:
{{begin-eqn}}
{{eqn |... | Let $m \in \Z: m > 1$.
Let $\struct {\Z_m, +_m}$ be the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $m$]].
Then every [[Definition:Subgroup|subgroup]] of $\struct {\Z_m, +_m}$ is an [[Definition:Ideal of Ring|ideal]] of the [[Definition:Ring of Integers Modulo m|ring of integer... | Let $H$ be a [[Definition:Subgroup|subgroup]] of $\struct {\Z_m, +_m}$
Suppose:
: $(1): \quad h + \ideal m \in H$, where $\ideal m$ is a [[Definition:Principal Ideal of Ring|principal ideal]] of $\struct {\Z_m, +_m, \times_m}$
and
: $(2): \quad n \in \N_{>0}$.
Then by definition of [[Definition:Integer Multiplicat... | Subgroup of Additive Group Modulo m is Ideal of Ring | https://proofwiki.org/wiki/Subgroup_of_Additive_Group_Modulo_m_is_Ideal_of_Ring | https://proofwiki.org/wiki/Subgroup_of_Additive_Group_Modulo_m_is_Ideal_of_Ring | [
"Subgroups",
"Ideal Theory",
"Additive Groups of Integers Modulo m",
"Modulo Addition"
] | [
"Definition:Additive Group of Integers Modulo m",
"Definition:Subgroup",
"Definition:Ideal of Ring",
"Definition:Ring of Integers Modulo m"
] | [
"Definition:Subgroup",
"Definition:Principal Ideal of Ring",
"Definition:Multiplication/Integers",
"Homomorphism of Powers/Integers",
"Definition:Quotient Mapping",
"Definition:Cyclic Group/Generator",
"Epimorphism from Integers to Cyclic Group"
] |
proofwiki-3108 | Bijection from Divisors to Subgroups of Cyclic Group | Let $G$ be a cyclic group of order $n$ generated by $a$.
Let $S = \set {m \in \Z_{>0}: m \divides n}$ be the set of all divisors of $n$.
Let $T$ be the set of all subgroups of $G$
Let $\phi: S \to T$ be the mapping defined as:
:$\phi: m \to \gen {a^{n / m} }$
where $\gen {a^{n / m} }$ is the subgroup generated by $a^{n... | From Subgroup of Finite Cyclic Group is Determined by Order, there exists exactly one subgroup $\gen {a^{n / m} }$ of $G$ with $a$ elements.
So the mapping as defined is indeed a bijection by definition.
{{qed}} | Let $G$ be a [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Structure|order]] $n$ [[Definition:Generator of Cyclic Group|generated by $a$]].
Let $S = \set {m \in \Z_{>0}: m \divides n}$ be the [[Definition:Set|set]] of all [[Definition:Divisor of Integer|divisors]] of $n$.
Let $T$ be the [[Definiti... | From [[Subgroup of Finite Cyclic Group is Determined by Order]], there exists exactly one [[Definition:Subgroup|subgroup]] $\gen {a^{n / m} }$ of $G$ with $a$ [[Definition:Element|elements]].
So the mapping as defined is indeed a [[Definition:Bijection|bijection]] by definition.
{{qed}} | Bijection from Divisors to Subgroups of Cyclic Group | https://proofwiki.org/wiki/Bijection_from_Divisors_to_Subgroups_of_Cyclic_Group | https://proofwiki.org/wiki/Bijection_from_Divisors_to_Subgroups_of_Cyclic_Group | [
"Subgroups",
"Cyclic Groups"
] | [
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Cyclic Group/Generator",
"Definition:Set",
"Definition:Divisor (Algebra)/Integer",
"Definition:Set",
"Definition:Subgroup",
"Definition:Mapping",
"Definition:Generator of Subgroup",
"Definition:Bijection"
] | [
"Subgroup of Finite Cyclic Group is Determined by Order",
"Definition:Subgroup",
"Definition:Element",
"Definition:Bijection"
] |
proofwiki-3109 | Generator of Additive Group Modulo m iff Unit of Ring | Let $m \in \Z: m > 1$.
Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$.
Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$.
Let $a \in \Z_m$.
Then:
:$a$ is a generator of $\struct {\Z_m, +_m}$
{{iff}}
:$a$ is a unit of $\struct {\Z_m, +_m, \times_m}$ | From Integers under Addition form Infinite Cyclic Group, the identity element $1_\Z$ of the ring of integers $\struct {\Z, +, \times}$ is a generator of the group $\struct {\Z, +}$.
Thus from Quotient Group of Cyclic Group, the identity element $1_{\Z_m}$ of the ring $\struct {\Z_m, +_m, \times_m}$ is a generator of th... | Let $m \in \Z: m > 1$.
Let $\struct {\Z_m, +_m}$ denote the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $m$]].
Let $\struct {\Z_m, +_m, \times_m}$ be the [[Definition:Ring of Integers Modulo m|ring of integers modulo $m$]].
Let $a \in \Z_m$.
Then:
:$a$ is a [[Definition:Gener... | From [[Integers under Addition form Infinite Cyclic Group]], the [[Definition:Identity Element|identity element]] $1_\Z$ of the [[Definition:Ring of Integers|ring of integers]] $\struct {\Z, +, \times}$ is a [[Definition:Generator of Cyclic Group|generator of the group]] $\struct {\Z, +}$.
Thus from [[Quotient Group o... | Generator of Additive Group Modulo m iff Unit of Ring | https://proofwiki.org/wiki/Generator_of_Additive_Group_Modulo_m_iff_Unit_of_Ring | https://proofwiki.org/wiki/Generator_of_Additive_Group_Modulo_m_iff_Unit_of_Ring | [
"Additive Groups of Integers Modulo m",
"Ring Theory"
] | [
"Definition:Additive Group of Integers Modulo m",
"Definition:Ring of Integers Modulo m",
"Definition:Cyclic Group/Generator",
"Definition:Unit of Ring"
] | [
"Integers under Addition form Infinite Cyclic Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring of Integers",
"Definition:Cyclic Group/Generator",
"Quotient Group of Cyclic Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring of Intege... |
proofwiki-3110 | Subspace of Real Differentiable Functions | Let $\mathbb J$ be an open interval of the real number line $\R$.
Let $\map \DD {\mathbb J}$ be the set of all differentiable real functions on $\mathbb J$.
Then $\struct {\map \DD {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$. | Note that by definition, $\map \DD {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map \DD {\mathbb J}$.
Let $\lambda \in \R$.
From Linear Combination of Derivatives, we have that:
:$f + \lambda g$ is differentiable on $\mathbb J$.
That is:
:$f + \lambda g \in \map \DD {\mathbb J}$
So, by One-Step Vector Sub... | Let $\mathbb J$ be an [[Definition:Open Real Interval|open interval]] of the [[Definition:Real Number Line|real number line]] $\R$.
Let $\map \DD {\mathbb J}$ be the set of all [[Definition:Differentiable on Interval|differentiable real functions]] on $\mathbb J$.
Then $\struct {\map \DD {\mathbb J}, +, \times}_\R$ ... | Note that by definition, $\map \DD {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map \DD {\mathbb J}$.
Let $\lambda \in \R$.
From [[Linear Combination of Derivatives]], we have that:
:$f + \lambda g$ is [[Definition:Differentiable Function|differentiable]] on $\mathbb J$.
That is:
:$f + \lambda g \in... | Subspace of Real Differentiable Functions | https://proofwiki.org/wiki/Subspace_of_Real_Differentiable_Functions | https://proofwiki.org/wiki/Subspace_of_Real_Differentiable_Functions | [
"Vector Subspaces",
"Differential Calculus"
] | [
"Definition:Real Interval/Open",
"Definition:Real Number/Real Number Line",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Vector Subspace",
"Definition:Vector Space"
] | [
"Linear Combination of Derivatives",
"Definition:Differentiable Mapping",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace"
] |
proofwiki-3111 | Subspace of Real Functions of Differentiability Class | Let $\mathbb J = \set {x \in \R: a < x < b}$ be an open interval of the real number line $\R$.
Let $\map {C^m} {\mathbb J}$ be the set of all continuous real functions on $\mathbb J$ in differentiability class $m$.
Then $\struct {\map {C^m} {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R... | Note that by definition, $\map {C^m} {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map {C^m} {\mathbb J}$.
Let $\lambda \in \R$.
Applying Linear Combination of Derivatives $m$ times we have:
:$f + \lambda g$ is $m$-times differentiable on $\mathbb J$ with $m$th derivative $f^{\paren m} + \lambda g^{\paren m}$.... | Let $\mathbb J = \set {x \in \R: a < x < b}$ be an [[Definition:Open Real Interval|open interval]] of the [[Definition:Real Number Line|real number line]] $\R$.
Let $\map {C^m} {\mathbb J}$ be the set of all [[Definition:Continuous Real Function|continuous real functions]] on $\mathbb J$ in [[Definition:Differentiabil... | Note that by definition, $\map {C^m} {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map {C^m} {\mathbb J}$.
Let $\lambda \in \R$.
Applying [[Linear Combination of Derivatives]] $m$ times we have:
:$f + \lambda g$ is $m$-times [[Definition:Differentiable Function|differentiable]] on $\mathbb J$ with [[Defini... | Subspace of Real Functions of Differentiability Class | https://proofwiki.org/wiki/Subspace_of_Real_Functions_of_Differentiability_Class | https://proofwiki.org/wiki/Subspace_of_Real_Functions_of_Differentiability_Class | [
"Vector Subspaces",
"Analysis"
] | [
"Definition:Real Interval/Open",
"Definition:Real Number/Real Number Line",
"Definition:Continuous Real Function",
"Definition:Differentiability Class",
"Definition:Vector Subspace",
"Definition:Vector Space"
] | [
"Linear Combination of Derivatives",
"Definition:Differentiable Mapping",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Differentiability Class",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Functions/Real/Combined Sum Rule",
"Definition:Continuous Re... |
proofwiki-3112 | Subspace of Riemann Integrable Functions | Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$.
Let $\map \RR {\mathbb J}$ be the set of all Riemann integrable functions on $\mathbb J$.
Then $\struct {\map \RR {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R... | Note that by definition, $\map \RR {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map \RR {\mathbb J}$.
Let $\lambda \in \R$.
By Linear Combination of Definite Integrals:
:$f + \lambda g$ is Riemann integrable on $\mathbb J$.
That is:
:$f + \lambda g \in \map \RR {\mathbb J}$
So by One-Step Vector Subspace Tes... | Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a [[Definition:Closed Real Interval|closed interval]] of the [[Definition:Real Number Line|real number line]] $\R$.
Let $\map \RR {\mathbb J}$ be the set of all [[Definition:Riemann Integrable Function|Riemann integrable functions]] on $\mathbb J$.
Then $\struct {\... | Note that by definition, $\map \RR {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map \RR {\mathbb J}$.
Let $\lambda \in \R$.
By [[Linear Combination of Definite Integrals]]:
:$f + \lambda g$ is [[Definition:Riemann Integrable Function|Riemann integrable]] on $\mathbb J$.
That is:
:$f + \lambda g \in \ma... | Subspace of Riemann Integrable Functions | https://proofwiki.org/wiki/Subspace_of_Riemann_Integrable_Functions | https://proofwiki.org/wiki/Subspace_of_Riemann_Integrable_Functions | [
"Vector Subspaces",
"Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number/Real Number Line",
"Definition:Definite Integral/Riemann",
"Definition:Vector Subspace",
"Definition:Vector Space"
] | [
"Linear Combination of Integrals/Definite",
"Definition:Definite Integral/Riemann",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace"
] |
proofwiki-3113 | Basis of Vector Space of Polynomial Functions | Let $B$ be the set of all the identity functions $I^n$ on $\R^n$ where $n \in \N^*$.
Then $B$ is a basis of the $\R$-vector space $\map P \R$ of all polynomial functions on $\R$. | By definition, every real polynomial function is a linear combination of $B$.
Suppose:
:$\ds \sum_{k \mathop = 0}^m \alpha_k I^k = 0, \alpha_m \ne 0$
Then by differentiating $m$ times, we obtain from Nth Derivative of Nth Power:
:$m! \alpha_m = 0$
whence $\alpha_m = 0$ which is a contradiction.
Hence $B$ is linearly in... | Let $B$ be the [[Definition:Set|set]] of all the [[Definition:Identity Mapping|identity functions]] $I^n$ on $\R^n$ where $n \in \N^*$.
Then $B$ is a [[Definition:Basis of Vector Space|basis]] of the [[Definition:Vector Space|$\R$-vector space]] $\map P \R$ of all [[Definition:Real Polynomial Function|polynomial func... | By definition, every [[Definition:Real Polynomial Function|real polynomial function]] is a [[Definition:Linear Combination|linear combination]] of $B$.
Suppose:
:$\ds \sum_{k \mathop = 0}^m \alpha_k I^k = 0, \alpha_m \ne 0$
Then by [[Definition:Differentiation|differentiating]] $m$ times, we obtain from [[Nth Deriva... | Basis of Vector Space of Polynomial Functions | https://proofwiki.org/wiki/Basis_of_Vector_Space_of_Polynomial_Functions | https://proofwiki.org/wiki/Basis_of_Vector_Space_of_Polynomial_Functions | [
"Bases of Vector Spaces",
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Basis of Vector Space",
"Definition:Vector Space",
"Definition:Polynomial Function/Real"
] | [
"Definition:Polynomial Function/Real",
"Definition:Linear Combination",
"Definition:Differentiation",
"Nth Derivative of Nth Power",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space"
] |
proofwiki-3114 | Ordered Basis for Coordinate Plane | Let $a_1, a_2 \in \R^2$ such that $\set {a_1, a_2}$ forms a linearly independent set.
Then $\tuple {a_1, a_2}$ is an ordered basis for the $\R$-vector space $\R^2$.
Hence the points on the plane can be uniquely identified by means of linear combinations of $a_1$ and $a_2$. | :500pxrightthumb
Let $P$ be any point in the plane for which we want to provide a linear combination of $a_1$ and $a_2$.
Let the distance from $O$ to the point determined by $a_1$ be defined as being $1$ unit of length on the line $L_1$.
Let the distance from $O$ to the point determined by $a_2$ be defined as being $1$... | Let $a_1, a_2 \in \R^2$ such that $\set {a_1, a_2}$ forms a [[Definition:Linearly Independent Set|linearly independent set]].
Then $\tuple {a_1, a_2}$ is an [[Definition:Ordered Basis|ordered basis]] for the [[Definition:Vector Space|$\R$-vector space]] $\R^2$.
Hence the points on the [[Definition:Plane|plane]] can ... | :[[File:OrderedBasisForPlane.png|500px|right|thumb]]
Let $P$ be any [[Definition:Point|point]] in the [[Definition:Plane|plane]] for which we want to provide a linear combination of $a_1$ and $a_2$.
Let the [[Definition:Length (Linear Measure)|distance]] from $O$ to the point determined by $a_1$ be defined as being ... | Ordered Basis for Coordinate Plane | https://proofwiki.org/wiki/Ordered_Basis_for_Coordinate_Plane | https://proofwiki.org/wiki/Ordered_Basis_for_Coordinate_Plane | [
"Coordinate Systems"
] | [
"Definition:Linearly Independent/Set",
"Definition:Ordered Basis",
"Definition:Vector Space",
"Definition:Plane Surface",
"Definition:Linear Combination"
] | [
"File:OrderedBasisForPlane.png",
"Definition:Point",
"Definition:Plane Surface",
"Definition:Linear Measure/Length",
"Definition:Line",
"Definition:Linear Measure/Length",
"Definition:Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Coordinate System/Coordinate",
"Parallelogram Law"
] |
proofwiki-3115 | Differentiation on Polynomials is Linear Operator | Let $\map P \R$ be the vector space of all polynomial functions on the real number line $\R$.
Then the differentiation operator $D$ on $\map P \R$ is a linear operator. | Let $\map f x, \map g x$ be real functions which are differentiable on $\R$.
Then from Linear Combination of Derivatives:
:$\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$
It follows from Real Polynomial Function is Differentiable that $\lambda D \map f x + \mu D \map g ... | Let $\map P \R$ be the [[Definition:Vector Space|vector space]] of all [[Definition:Real Polynomial Function|polynomial functions]] on the [[Definition:Real Number Line|real number line]] $\R$.
Then the [[Definition:Differentiation|differentiation]] operator $D$ on $\map P \R$ is a [[Definition:Linear Operator|linear... | Let $\map f x, \map g x$ be [[Definition:Real Function|real functions]] which are [[Definition:Differentiable on Interval|differentiable]] on $\R$.
Then from [[Linear Combination of Derivatives]]:
:$\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$
It follows from [[Real... | Differentiation on Polynomials is Linear Operator | https://proofwiki.org/wiki/Differentiation_on_Polynomials_is_Linear_Operator | https://proofwiki.org/wiki/Differentiation_on_Polynomials_is_Linear_Operator | [
"Linear Operators",
"Differential Calculus",
"Real Polynomial Functions"
] | [
"Definition:Vector Space",
"Definition:Polynomial Function/Real",
"Definition:Real Number/Real Number Line",
"Definition:Differentiation",
"Definition:Linear Operator"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Linear Combination of Derivatives",
"Real Polynomial Function is Differentiable",
"Definition:Differentiable Mapping/Real Function/Interval",
"Condition for Linear Transformation"
] |
proofwiki-3116 | Integration on Polynomials is Linear Operator | Let $\map P \R$ be the vector space of all polynomial functions on the real number line $\R$.
Let $S$ be the mapping defined as:
:$\ds \forall p \in \map P \R: \forall x \in \R: \map S {\map p x} = \int_0^x \map p t \rd t$
Then $S$ is a linear operator on $\map P \R$. | Let $\map f x, \map g x$ be real functions which are integrable on $\R$.
Let $\closedint a b$ be a closed interval of $\R$.
Then from Linear Combination of Definite Integrals, $\lambda f + \mu g$ is integrable on $\closedint a b$ and:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map... | Let $\map P \R$ be the [[Definition:Vector Space|vector space]] of all [[Definition:Real Polynomial Function|polynomial functions]] on the [[Definition:Real Number Line|real number line]] $\R$.
Let $S$ be the [[Definition:Mapping|mapping]] defined as:
:$\ds \forall p \in \map P \R: \forall x \in \R: \map S {\map p x} ... | Let $\map f x, \map g x$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]] on $\R$.
Let $\closedint a b$ be a [[Definition:Closed Interval|closed interval]] of $\R$.
Then from [[Linear Combination of Definite Integrals]], $\lambda f + \mu g$ is [[Definition:Integra... | Integration on Polynomials is Linear Operator | https://proofwiki.org/wiki/Integration_on_Polynomials_is_Linear_Operator | https://proofwiki.org/wiki/Integration_on_Polynomials_is_Linear_Operator | [
"Linear Operators",
"Integral Calculus",
"Real Polynomial Functions"
] | [
"Definition:Vector Space",
"Definition:Polynomial Function/Real",
"Definition:Real Number/Real Number Line",
"Definition:Mapping",
"Definition:Linear Operator"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Interval/Ordered Set/Closed",
"Linear Combination of Integrals/Definite",
"Definition:Integrable Function",
"Real Polynomial Function is Continuous",
"Definition:Continuous Real Function",
"Continuous Real Function is Darboux In... |
proofwiki-3117 | Noether's Theorem (Calculus of Variations) | Let $y_i$, $F$, $\Psi_i$, $\Phi$ be real functions.
Let $x, \epsilon \in \R$.
Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ and $\mathbf \Psi = \sequence{\Psi_i}_{1 \mathop \le i \mathop \le n}$ be vectors.
Let
:$\Phi = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}, \quad \Psi_i = \map {\Psi_i} {x... | Apply Taylor's Theorem to the transformations $X$, $\mathbf Y$ at the point $\epsilon = 0$:
{{begin-eqn}}
{{eqn | l = X
| r = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}
}}
{{eqn | r = \map \Phi {x, \mathbf y, \mathbf y'; 0} + \valueat {\epsilon \frac {\partial \map \Phi {x, \mathbf y, \mathbf y'; \epsilon} } ... | Let $y_i$, $F$, $\Psi_i$, $\Phi$ be real functions.
Let $x, \epsilon \in \R$.
Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ and $\mathbf \Psi = \sequence{\Psi_i}_{1 \mathop \le i \mathop \le n}$ be [[Definition:Vector (Linear Algebra)|vectors]].
Let
:$\Phi = \map \Phi {x, \mathbf y, \mathbf y'; ... | Apply [[Taylor's Theorem]] to the transformations $X$, $\mathbf Y$ at the point $\epsilon = 0$:
{{begin-eqn}}
{{eqn | l = X
| r = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}
}}
{{eqn | r = \map \Phi {x, \mathbf y, \mathbf y'; 0} + \valueat {\epsilon \frac {\partial \map \Phi {x, \mathbf y, \mathbf y'; \epsilo... | Noether's Theorem (Calculus of Variations) | https://proofwiki.org/wiki/Noether's_Theorem_(Calculus_of_Variations) | https://proofwiki.org/wiki/Noether's_Theorem_(Calculus_of_Variations) | [
"Partial Differential Equations",
"Calculus of Variations"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Variable",
"Definition:Functional/Real",
"Definition:Invariant Functional under Transformation",
"Definition:Constant"
] | [
"Taylor's Theorem",
"General Variation of Integral Functional/Dependent on N Functions",
"Definition:Invariant Functional under Transformation",
"Definition:Differential of Mapping/Functional",
"Definition:Constant Mapping",
"Definition:Constant"
] |
proofwiki-3118 | Transpose of Linear Transformation is a Linear Transformation | Let $R$ be a commutative ring.
Let $G$ and $H$ be $R$-modules.
Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively.
Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.
Let $u \in \map {\LL_R} {G, H}$.
Let $u^\intercal: H^* \to G^*$ be the transpose of $u$.
Then $u^\int... | By definition of evaluation linear transformation:
:$\forall x \in G: y \in H^*: \innerprod x {\map {u^\intercal} y} = \innerprod {\map u x} y$
Since we have:
{{begin-eqn}}
{{eqn | l = \innerprod x {\map {u^\intercal} {y + z} }
| r = \innerprod {\map u x} {y + z}
| c =
}}
{{eqn | r = \innerprod {\map u x} ... | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $G$ and $H$ be [[Definition:Module over Ring|$R$-modules]].
Let $G^*$ and $H^*$ be the [[Definition:Algebraic Dual|algebraic duals]] of $G$ and $H$ respectively.
Let $\map {\LL_R} {G, H}$ be [[Definition:Set of All Linear Transformations|the set of ... | By definition of [[Definition:Evaluation Linear Transformation/Module Theory|evaluation linear transformation]]:
:$\forall x \in G: y \in H^*: \innerprod x {\map {u^\intercal} y} = \innerprod {\map u x} y$
Since we have:
{{begin-eqn}}
{{eqn | l = \innerprod x {\map {u^\intercal} {y + z} }
| r = \innerprod {\map... | Transpose of Linear Transformation is a Linear Transformation | https://proofwiki.org/wiki/Transpose_of_Linear_Transformation_is_a_Linear_Transformation | https://proofwiki.org/wiki/Transpose_of_Linear_Transformation_is_a_Linear_Transformation | [
"Linear Transformations"
] | [
"Definition:Commutative Ring",
"Definition:Module over Ring",
"Definition:Algebraic Dual",
"Definition:Set of All Linear Transformations",
"Definition:Transpose of Linear Transformation",
"Definition:Linear Transformation"
] | [
"Definition:Evaluation Linear Transformation/Module Theory",
"Definition:Linear Transformation"
] |
proofwiki-3119 | Equivalence of Definitions of Supremum of Real-Valued Function | Let $S \subseteq \R$ be a subset of the real numbers.
Let $f: S \to \R$ be a real function on $S$.
{{TFAE|def = Supremum of Real-Valued Function}} | === Definition 1 implies Definition 2 ===
Let $K \in \R$ be a supremum of $f$ by definition 1.
Then from the definition:
:$\text{(a)}: \quad K$ is an upper bound of $\map f x$ in $\R$.
:$\text{(b)}: \quad K \le M$ for all upper bounds $M$ of $f \sqbrk S$ in $\R$.
As $K$ is an upper bound it follows that:
:$(1): \quad \... | Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $f: S \to \R$ be a [[Definition:Real Function|real function]] on $S$.
{{TFAE|def = Supremum of Real-Valued Function}} | === Definition 1 implies Definition 2 ===
Let $K \in \R$ be a [[Definition:Supremum of Mapping/Real-Valued Function/Definition 1|supremum of $f$ by definition 1]].
Then from the definition:
:$\text{(a)}: \quad K$ is an [[Definition:Upper Bound of Real-Valued Function|upper bound]] of $\map f x$ in $\R$.
:$\text{(b)... | Equivalence of Definitions of Supremum of Real-Valued Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Supremum_of_Real-Valued_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Supremum_of_Real-Valued_Function | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Function"
] | [
"Definition:Supremum of Mapping/Real-Valued Function/Definition 1",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Upper Bound of Mapping/Real-Valued",
"Real Plus Epsilon",
"Proof by Contradic... |
proofwiki-3120 | Equivalence of Definitions of Infimum of Real-Valued Function | Let $S \subseteq \R$ be a subset of the real numbers.
Let $f: S \to \R$ be a real function on $S$.
{{TFAE|def = Infimum of Real-Valued Function}} | {{wtd|Use Characterizing Property of Infimum of Subset of Real Numbers}} | Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $f: S \to \R$ be a [[Definition:Real Function|real function]] on $S$.
{{TFAE|def = Infimum of Real-Valued Function}} | {{wtd|Use [[Characterizing Property of Infimum of Subset of Real Numbers]]}} | Equivalence of Definitions of Infimum of Real-Valued Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Infimum_of_Real-Valued_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Infimum_of_Real-Valued_Function | [
"Infima"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Function"
] | [
"Characterizing Property of Infimum of Subset of Real Numbers"
] |
proofwiki-3121 | Supremum does not Precede Infimum | Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ admit both a supremum $M$ and an infimum $m$.
Then $m \preceq M$. | By definition of supremum:
:$\forall a \in T: a \preceq M$
By definition of infimum:
:$\forall a \in T: m \preceq a$
The result follows from transitivity of ordering.
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T \subseteq S$ admit both a [[Definition:Supremum of Set|supremum]] $M$ and an [[Definition:Infimum of Set|infimum]] $m$.
Then $m \preceq M$. | By definition of [[Definition:Supremum of Set|supremum]]:
:$\forall a \in T: a \preceq M$
By definition of [[Definition:Infimum of Set|infimum]]:
:$\forall a \in T: m \preceq a$
The result follows from [[Definition:Transitive Relation|transitivity]] of [[Definition:Ordering|ordering]].
{{qed}} | Supremum does not Precede Infimum | https://proofwiki.org/wiki/Supremum_does_not_Precede_Infimum | https://proofwiki.org/wiki/Supremum_does_not_Precede_Infimum | [
"Suprema",
"Infima"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Infimum of Set"
] | [
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Transitive Relation",
"Definition:Ordering"
] |
proofwiki-3122 | Integer Multiples form Commutative Ring | Let $n \Z$ be the set of integer multiples of $n$.
Then $\struct {n \Z, +, \times}$ is a commutative ring.
Unless $n = 1$, $\struct {n \Z, +, \times}$ is not a ring with unity. | From Integer Multiples under Addition form Infinite Cyclic Group, $\struct {n \Z, +}$ is a cyclic group
From Cyclic Group is Abelian, $\struct {n \Z, +}$ is abelian.
From Integer Multiples Closed under Multiplication and Integer Multiplication is Associative, we have that $\struct {n \Z, \times}$ is a semigroup.
From I... | Let $n \Z$ be the [[Definition:Set of Integer Multiples|set of integer multiples]] of $n$.
Then $\struct {n \Z, +, \times}$ is a [[Definition:Commutative Ring|commutative ring]].
Unless $n = 1$, $\struct {n \Z, +, \times}$ is not a [[Definition:Ring with Unity|ring with unity]]. | From [[Integer Multiples under Addition form Infinite Cyclic Group]], $\struct {n \Z, +}$ is a [[Definition:Cyclic Group|cyclic group]]
From [[Cyclic Group is Abelian]], $\struct {n \Z, +}$ is [[Definition:Abelian Group|abelian]].
From [[Integer Multiples Closed under Multiplication]] and [[Integer Multiplication is ... | Integer Multiples form Commutative Ring | https://proofwiki.org/wiki/Integer_Multiples_form_Commutative_Ring | https://proofwiki.org/wiki/Integer_Multiples_form_Commutative_Ring | [
"Integers",
"Commutative Rings",
"Integer Multiples form Commutative Ring"
] | [
"Definition:Set of Integer Multiples",
"Definition:Commutative Ring",
"Definition:Ring with Unity"
] | [
"Integer Multiples under Addition form Infinite Cyclic Group",
"Definition:Cyclic Group",
"Cyclic Group is Abelian",
"Definition:Abelian Group",
"Integer Multiples Closed under Multiplication",
"Integer Multiplication is Associative",
"Definition:Semigroup",
"Integer Multiplication Distributes over Ad... |
proofwiki-3123 | Quaternions Defined by Matrices | Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\MM_\C} 2$:
:$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
\qquad
\... | This is demonstrated by straightforward application of conventional matrix multiplication:
{{begin-eqn}}
{{eqn | l = \mathbf i \mathbf j
| r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
| c =
}}
{{eqn | r = \begin{bmatrix} i \cdot 0 + 0 \cdot -1 & i \cdot 1 ... | Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four [[Definition:Element|elements]] of the [[Definition:Matrix Space|matrix space]] $\map {\MM_\C} 2$:
:$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf j = \b... | This is demonstrated by straightforward application of [[Definition:Matrix Product (Conventional)|conventional matrix multiplication]]:
{{begin-eqn}}
{{eqn | l = \mathbf i \mathbf j
| r = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
| c =
}}
{{eqn | r = \beg... | Quaternions Defined by Matrices | https://proofwiki.org/wiki/Quaternions_Defined_by_Matrices | https://proofwiki.org/wiki/Quaternions_Defined_by_Matrices | [
"Matrix Algebra",
"Quaternion Group"
] | [
"Definition:Element",
"Definition:Matrix Space",
"Definition:Complex Number"
] | [
"Definition:Matrix Product (Conventional)",
"Matrix Multiplication is Associative"
] |
proofwiki-3124 | Matrix Form of Quaternion | Let $\mathbf x$ be a quaternion such that:
:$\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
When the quaternion basis is expressed in the form of matrices:
:<nowiki>$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\math... | {{begin-eqn}}
{{eqn | l = \mathbf x
| r = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k
| c =
}}
{{eqn | r = a \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} + c \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & i \\ i & 0 \end{bma... | Let $\mathbf x$ be a [[Definition:Quaternion|quaternion]] such that:
:$\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
When the [[Definition:Quaternion|quaternion]] basis is [[Quaternions Defined by Matrices|expressed in the form of matrices]]:
:<nowiki>$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \... | {{begin-eqn}}
{{eqn | l = \mathbf x
| r = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k
| c =
}}
{{eqn | r = a \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} + c \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & i \\ i & 0 \end{bma... | Matrix Form of Quaternion | https://proofwiki.org/wiki/Matrix_Form_of_Quaternion | https://proofwiki.org/wiki/Matrix_Form_of_Quaternion | [
"Quaternions",
"Matrix Algebra"
] | [
"Definition:Quaternion",
"Definition:Quaternion",
"Quaternions Defined by Matrices",
"Definition:Quaternion",
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [] |
proofwiki-3125 | Quaternion Multiplication | Let $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ be quaternions.
Then their '''product''' is given by:
{{begin-eqn}}
{{eqn | ll=\mathbf x_1 \mathbf x_2
| l==
| o=
| r=\left({a_1 a_2 - b_1... | From Matrix Form of Quaternion we have that:
:$\mathbf x_1 \mathbf x_2 = \begin{bmatrix} a_1 + b_1 i & c_1 + d_1 i \\ -c_1 + d_1 i & a_1 - b_1 i \end{bmatrix} \begin{bmatrix} a_2 + b_2 i & c_2 + d_2 i \\ -c_2 + d_2 i & a_2 - b_2 i \end{bmatrix}$
Let $\mathbf x_1 \mathbf x_2 = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} &... | Let $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ be [[Definition:Quaternion|quaternions]].
Then their '''product''' is given by:
{{begin-eqn}}
{{eqn | ll=\mathbf x_1 \mathbf x_2
| l==
| o=
... | From [[Matrix Form of Quaternion]] we have that:
:$\mathbf x_1 \mathbf x_2 = \begin{bmatrix} a_1 + b_1 i & c_1 + d_1 i \\ -c_1 + d_1 i & a_1 - b_1 i \end{bmatrix} \begin{bmatrix} a_2 + b_2 i & c_2 + d_2 i \\ -c_2 + d_2 i & a_2 - b_2 i \end{bmatrix}$
Let $\mathbf x_1 \mathbf x_2 = \begin{bmatrix} p_{11} & p_{12} \\ p_... | Quaternion Multiplication | https://proofwiki.org/wiki/Quaternion_Multiplication | https://proofwiki.org/wiki/Quaternion_Multiplication | [
"Quaternions"
] | [
"Definition:Quaternion"
] | [
"Matrix Form of Quaternion",
"Definition:Matrix Product (Conventional)",
"Matrix Form of Quaternion",
"Category:Quaternions"
] |
proofwiki-3126 | Quaternion Addition forms Abelian Group | Let $\mathbb H$ be the set of quaternions.
Then $\struct {\mathbb H, +}$, where $+$ denotes quaternion addition, is an abelian group. | Taking the abelian group axioms in turn: | Let $\mathbb H$ be the [[Definition:Set|set]] of [[Definition:Quaternion|quaternions]].
Then $\struct {\mathbb H, +}$, where $+$ denotes [[Definition:Quaternion Addition|quaternion addition]], is an [[Definition:Abelian Group|abelian group]]. | Taking the [[Axiom:Abelian Group Axioms|abelian group axioms]] in turn: | Quaternion Addition forms Abelian Group | https://proofwiki.org/wiki/Quaternion_Addition_forms_Abelian_Group | https://proofwiki.org/wiki/Quaternion_Addition_forms_Abelian_Group | [
"Quaternions",
"Abelian Groups"
] | [
"Definition:Set",
"Definition:Quaternion",
"Definition:Quaternion/Addition",
"Definition:Abelian Group"
] | [
"Axiom:Abelian Group Axioms",
"Axiom:Abelian Group Axioms"
] |
proofwiki-3127 | Ring of Quaternions is Ring | The set $\mathbb H$ of quaternions forms a ring under the operations of addition and multiplication. | From Quaternion Addition forms Abelian Group, $\mathbb H$ forms an abelian group under quaternion addition.
From the definition it is clear that quaternion multiplication is closed.
We have from Matrix Form of Quaternion that quaternions can be expressed in matrix form.
From Quaternion Multiplication we have that quate... | The [[Definition:Set|set]] $\mathbb H$ of [[Definition:Quaternion|quaternions]] forms a [[Definition:Ring (Abstract Algebra)|ring]] under the [[Definition:Binary Operation|operations]] of [[Definition:Quaternion Addition|addition]] and [[Definition:Quaternion Multiplication|multiplication]]. | From [[Quaternion Addition forms Abelian Group]], $\mathbb H$ forms an [[Definition:Abelian Group|abelian group]] under [[Definition:Quaternion Addition|quaternion addition]].
From the definition it is clear that [[Definition:Quaternion Multiplication|quaternion multiplication]] is [[Definition:Closure (Abstract Alge... | Ring of Quaternions is Ring | https://proofwiki.org/wiki/Ring_of_Quaternions_is_Ring | https://proofwiki.org/wiki/Ring_of_Quaternions_is_Ring | [
"Quaternions",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Quaternion",
"Definition:Ring (Abstract Algebra)",
"Definition:Operation/Binary Operation",
"Definition:Quaternion/Addition",
"Definition:Quaternion/Multiplication"
] | [
"Quaternion Addition forms Abelian Group",
"Definition:Abelian Group",
"Definition:Quaternion/Addition",
"Definition:Quaternion/Multiplication",
"Definition:Closure (Abstract Algebra)",
"Matrix Form of Quaternion",
"Definition:Quaternion",
"Definition:Matrix",
"Quaternion Multiplication",
"Definit... |
proofwiki-3128 | Quaternions Subring of Complex Matrix Space | The Ring of Quaternions is a subring of the matrix space $\map {\MM_\C} 2$. | From Matrix Form of Quaternion it is clear that the quaternions $\H$ can be expressed in matrix form, as elements of $\map {\MM_\C} 2$.
Thus $\H \subseteq \map {\MM_\C} 2$.
As the quaternions form a ring, the result follows by definition of subring.
{{qed}} | The [[Ring of Quaternions]] is a [[Definition:Subring|subring]] of the [[Definition:Matrix Space|matrix space]] $\map {\MM_\C} 2$. | From [[Matrix Form of Quaternion]] it is clear that the [[Definition:Quaternion|quaternions]] $\H$ can be expressed in [[Definition:Matrix|matrix]] form, as elements of $\map {\MM_\C} 2$.
Thus $\H \subseteq \map {\MM_\C} 2$.
As [[Ring of Quaternions|the quaternions form a ring]], the result follows by definition of [... | Quaternions Subring of Complex Matrix Space | https://proofwiki.org/wiki/Quaternions_Subring_of_Complex_Matrix_Space | https://proofwiki.org/wiki/Quaternions_Subring_of_Complex_Matrix_Space | [
"Quaternions",
"Matrix Algebra"
] | [
"Ring of Quaternions is Ring",
"Definition:Subring",
"Definition:Matrix Space"
] | [
"Matrix Form of Quaternion",
"Definition:Quaternion",
"Definition:Matrix",
"Ring of Quaternions is Ring",
"Definition:Subring"
] |
proofwiki-3129 | Product of Quaternion with Conjugate | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
Then their product is given by:
:$\mathbf x \overline {\mathbf x} = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1 = \overline {\mathbf x} \mathbf x$ | From the definition of quaternion multiplication:
{{begin-eqn}}
{{eqn | l = \mathbf x \overline {\mathbf x}
| r = \paren {a^2 - b \paren {-b} - c \paren {-c} - d \paren {-d} } \mathbf 1
| c =
}}
{{eqn | o =
| ro= +
| r = \paren {a \paren {-b} + b a + c \paren {-d} - d \paren {-c} } \mathbf i
... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\overline {\mathbf x}$ be the [[Definition:Conjugate Quaternion|conjugate]] of $\mathbf x$.
Then their [[Definition:Quaternion Multiplication|product]] is given by:
:$\mathbf x \overline {\mathbf x... | From the definition of [[Definition:Quaternion Multiplication|quaternion multiplication]]:
{{begin-eqn}}
{{eqn | l = \mathbf x \overline {\mathbf x}
| r = \paren {a^2 - b \paren {-b} - c \paren {-c} - d \paren {-d} } \mathbf 1
| c =
}}
{{eqn | o =
| ro= +
| r = \paren {a \paren {-b} + b a + c... | Product of Quaternion with Conjugate | https://proofwiki.org/wiki/Product_of_Quaternion_with_Conjugate | https://proofwiki.org/wiki/Product_of_Quaternion_with_Conjugate | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Quaternion/Multiplication"
] | [
"Definition:Quaternion/Multiplication",
"Real Multiplication is Commutative",
"Definition:Commutative/Elements"
] |
proofwiki-3130 | Multiplicative Inverse of Quaternion | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion such that $\mathbf x \ne \mathbf 0$.
Then $\mathbf x$ has an inverse $\mathbf x^{-1}$ under the operation of quaternion multiplication:
:$\mathbf x^{-1} = \lambda \overline {\mathbf x}$
where:
:$\lambda = \dfrac 1 {a^2 + b^2 + c^2 +... | From Multiplicative Identity for Quaternions, we need to show that $\lambda \overline{\mathbf x} \mathbf x = \mathbf 1 = \mathbf x \lambda \overline{\mathbf x}$.
From Product of Quaternion with Conjugate we have that:
:$\overline{\mathbf x} \mathbf x = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1$
Using the definition of $... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]] such that $\mathbf x \ne \mathbf 0$.
Then $\mathbf x$ has an [[Definition:Inverse Element|inverse]] $\mathbf x^{-1}$ under the operation of [[Definition:Quaternion Multiplication|quaternion multiplication... | From [[Multiplicative Identity for Quaternions]], we need to show that $\lambda \overline{\mathbf x} \mathbf x = \mathbf 1 = \mathbf x \lambda \overline{\mathbf x}$.
From [[Product of Quaternion with Conjugate]] we have that:
:$\overline{\mathbf x} \mathbf x = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1$
Using the de... | Multiplicative Inverse of Quaternion | https://proofwiki.org/wiki/Multiplicative_Inverse_of_Quaternion | https://proofwiki.org/wiki/Multiplicative_Inverse_of_Quaternion | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Quaternion/Multiplication"
] | [
"Multiplicative Identity for Quaternions",
"Product of Quaternion with Conjugate",
"Definition:Quaternion",
"Left Inverse for All is Right Inverse"
] |
proofwiki-3131 | Multiplicative Identity for Quaternions | In the set of quaternions $\mathbb H$, the element:
:$\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$
serves as the identity element for quaternion multiplication.
This element is written $\mathbf 1$. | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.
From the definition of quaternion multiplication:
{{begin-eqn}}
{{eqn | l = \mathbf x \mathbf 1
| r = \paren {a \cdot 1 - b \cdot 0 - c \cdot 0 - d \cdot 0} \mathbf 1
| c =
}}
{{eqn | o = +
| r = \paren {a \cdot 0 + b \cdot 1 + ... | In the [[Definition:Set|set]] of [[Definition:Quaternion|quaternions]] $\mathbb H$, the [[Definition:Element|element]]:
:$\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$
serves as the [[Definition:Identity Element|identity element]] for [[Definition:Quaternion Multiplication|quaternion multiplication]].
This [[D... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.
From the definition of [[Definition:Quaternion Multiplication|quaternion multiplication]]:
{{begin-eqn}}
{{eqn | l = \mathbf x \mathbf 1
| r = \paren {a \cdot 1 - b \cdot 0 - c \cdot 0 - d \cdot 0} \mathbf 1
| c =
}}
{{eqn | o = +
... | Multiplicative Identity for Quaternions | https://proofwiki.org/wiki/Multiplicative_Identity_for_Quaternions | https://proofwiki.org/wiki/Multiplicative_Identity_for_Quaternions | [
"Quaternions"
] | [
"Definition:Set",
"Definition:Quaternion",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Quaternion/Multiplication",
"Definition:Element"
] | [
"Definition:Quaternion/Multiplication",
"Category:Quaternions"
] |
proofwiki-3132 | Quaternions form Skew Field | The set $\H$ of quaternions forms a '''skew field''' under the operations of addition and multiplication. | From Ring of Quaternions is Ring we have that $\H$ forms a ring.
From Multiplicative Identity for Quaternions we have that $\mathbf 1$ is the identity for quaternion multiplication.
From Multiplicative Inverse of Quaternion we have that every element of $\H$ except $\mathbf 0$ has an inverse under quaternion multiplica... | The [[Definition:Set|set]] $\H$ of [[Definition:Quaternion|quaternions]] forms a '''[[Definition:Skew Field|skew field]]''' under the [[Definition:Binary Operation|operations]] of [[Definition:Quaternion Addition|addition]] and [[Definition:Quaternion Multiplication|multiplication]]. | From [[Ring of Quaternions is Ring]] we have that $\H$ forms a [[Definition:Ring (Abstract Algebra)|ring]].
From [[Multiplicative Identity for Quaternions]] we have that $\mathbf 1$ is the [[Definition:Identity Element|identity]] for [[Definition:Quaternion Multiplication|quaternion multiplication]].
From [[Multiplic... | Quaternions form Skew Field | https://proofwiki.org/wiki/Quaternions_form_Skew_Field | https://proofwiki.org/wiki/Quaternions_form_Skew_Field | [
"Quaternions",
"Examples of Skew Fields"
] | [
"Definition:Set",
"Definition:Quaternion",
"Definition:Skew Field",
"Definition:Operation/Binary Operation",
"Definition:Quaternion/Addition",
"Definition:Quaternion/Multiplication"
] | [
"Ring of Quaternions is Ring",
"Definition:Ring (Abstract Algebra)",
"Multiplicative Identity for Quaternions",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Quaternion/Multiplication",
"Multiplicative Inverse of Quaternion",
"Definition:Inverse (Abstract Algebra)/Inverse",
... |
proofwiki-3133 | Cancellation Law for Ring Product of Integral Domain | Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.
Let $D^*$ denote $D \setminus \set {0_D}$, that is, $D$ without its zero.
Let $a \in D^*$.
Then:
:$\forall x, y \in D: a \circ x = a \circ y \implies x = y$
That is, all elements of $D^*$ are cancellable for the ring product. | From the definition of integral domain, no elements of $D^*$ are zero divisors.
From Ring Element is Zero Divisor iff not Cancellable, it follows that all elements of $D^*$ are cancellable for the ring product $\circ$.
{{qed}} | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Ring Zero|zero]] is $0_D$.
Let $D^*$ denote $D \setminus \set {0_D}$, that is, $D$ without its [[Definition:Ring Zero|zero]].
Let $a \in D^*$.
Then:
:$\forall x, y \in D: a \circ x = a \circ y \implies x = y$
That... | From the definition of [[Definition:Integral Domain|integral domain]], no elements of $D^*$ are [[Definition:Zero Divisor of Ring|zero divisors]].
From [[Ring Element is Zero Divisor iff not Cancellable]], it follows that all elements of $D^*$ are [[Definition:Cancellable Element|cancellable]] for the [[Definition:Rin... | Cancellation Law for Ring Product of Integral Domain | https://proofwiki.org/wiki/Cancellation_Law_for_Ring_Product_of_Integral_Domain | https://proofwiki.org/wiki/Cancellation_Law_for_Ring_Product_of_Integral_Domain | [
"Integral Domains",
"Cancellation Laws"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Cancellable Element",
"Definition:Ring (Abstract Algebra)/Product"
] | [
"Definition:Integral Domain",
"Definition:Zero Divisor/Ring",
"Ring Element is Zero Divisor iff not Cancellable",
"Definition:Cancellable Element",
"Definition:Ring (Abstract Algebra)/Product"
] |
proofwiki-3134 | Gaussian Integers form Integral Domain | The ring of Gaussian integers $\struct {\Z \sqbrk i, +, \times}$ is an integral domain. | The set of complex numbers $\C$ forms a field, which is by definition a division ring.
We have that $\Z \sqbrk i \subset \C$.
So from Cancellable Element is Cancellable in Subset, all non-zero elements of $\Z \sqbrk i$ are cancellable for complex multiplication.
The identity element for complex multiplication is $1 + 0... | The [[Definition:Ring of Gaussian Integers|ring of Gaussian integers]] $\struct {\Z \sqbrk i, +, \times}$ is an [[Definition:Integral Domain|integral domain]]. | The [[Complex Numbers form Field|set of complex numbers $\C$ forms a field]], which is [[Definition:Field (Abstract Algebra)|by definition]] a [[Definition:Division Ring|division ring]].
We have that $\Z \sqbrk i \subset \C$.
So from [[Cancellable Element is Cancellable in Subset]], all non-zero elements of $\Z \sqbr... | Gaussian Integers form Integral Domain | https://proofwiki.org/wiki/Gaussian_Integers_form_Integral_Domain | https://proofwiki.org/wiki/Gaussian_Integers_form_Integral_Domain | [
"Examples of Integral Domains",
"Gaussian Integers"
] | [
"Definition:Ring of Gaussian Integers",
"Definition:Integral Domain"
] | [
"Complex Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Division Ring",
"Cancellable Element is Cancellable in Subset",
"Definition:Cancellable Element",
"Definition:Multiplication/Complex Numbers",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multi... |
proofwiki-3135 | Set Intersection Not Cancellable | Let $S$ be a set and let $\powerset S$ be the power set of $S$.
Let $S_1, S_2, T \in \powerset S$.
Suppose that $S_1 \cap T = S_2 \cap T$.
Then it is not necessarily the case that $S_1 = S_2$. | Proof by Counterexample:
Let $S = \set {1, 2, 3}$.
Let $T = \set 3$.
Let $S_1 = \set {1, 3}, S_2 = \set {2, 3}$
Then $S_1 \cap T = S_2 \cap T = \set 3$ but $S_1 \ne S_2$.
{{qed}}
Category:Set Intersection
brmjm7pmkl21sk8q4nujb4mkne5r398 | Let $S$ be a [[Definition:Set|set]] and let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Let $S_1, S_2, T \in \powerset S$.
Suppose that $S_1 \cap T = S_2 \cap T$.
Then it is not necessarily the case that $S_1 = S_2$. | [[Proof by Counterexample]]:
Let $S = \set {1, 2, 3}$.
Let $T = \set 3$.
Let $S_1 = \set {1, 3}, S_2 = \set {2, 3}$
Then $S_1 \cap T = S_2 \cap T = \set 3$ but $S_1 \ne S_2$.
{{qed}}
[[Category:Set Intersection]]
brmjm7pmkl21sk8q4nujb4mkne5r398 | Set Intersection Not Cancellable | https://proofwiki.org/wiki/Set_Intersection_Not_Cancellable | https://proofwiki.org/wiki/Set_Intersection_Not_Cancellable | [
"Set Intersection"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Proof by Counterexample",
"Category:Set Intersection"
] |
proofwiki-3136 | Sum of Triangular Matrices | Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be square matrices of order $n$.
Let $\mathbf C = \mathbf A + \mathbf B$ be the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, then so is $\mathbf C$.
If $\mathbf A$ and $\mathbf ... | From the definition of matrix addition, we have:
:$\forall i, j \in \left[{1 .. n}\right]: c_{ij} = a_{ij} + b_{ij}$
If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, we have:
:$\forall i > j: a_{ij} = b_{ij} = 0$
Hence:
:$\forall i > j: c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0$
and so $\mathbf C$ is itself upp... | Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be [[Definition:Square Matrix|square matrices]] of order $n$.
Let $\mathbf C = \mathbf A + \mathbf B$ be the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of $\mathbf A$ and $\mathbf B$.
If $\mathbf A$ and $\mathbf B$ are [[Defi... | From the definition of [[Definition:Matrix Entrywise Addition|matrix addition]], we have:
:$\forall i, j \in \left[{1 .. n}\right]: c_{ij} = a_{ij} + b_{ij}$
If $\mathbf A$ and $\mathbf B$ are [[Definition:Upper Triangular Matrix|upper triangular matrices]], we have:
:$\forall i > j: a_{ij} = b_{ij} = 0$
Hence:
:$\... | Sum of Triangular Matrices | https://proofwiki.org/wiki/Sum_of_Triangular_Matrices | https://proofwiki.org/wiki/Sum_of_Triangular_Matrices | [
"Triangular Matrices",
"Matrix Entrywise Addition"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix Entrywise Addition",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Category:Triangular Matrices",
"Categor... |
proofwiki-3137 | Negative of Triangular Matrix | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $-\mathbf A$ be the negative of $\mathbf A$.
If $\mathbf A$ is an upper triangular matrix, then so is $-\mathbf A$.
If $\mathbf A$ is a lower triangular matrix, then so is $-\mathbf A$. | From the definition of negative matrix, we have:
:$\forall i, j \in \closedint 1 n: \sqbrk {-a}_{i j} = -a_{i j}$
If $\mathbf A$ is an upper triangular matrix, we have:
:$\forall i > j: a_{i j} = 0$
Hence:
:$\forall i > j: \sqbrk {-a}_{i j} = -a_{i j} = 0$
and so $-\mathbf A$ is itself upper triangular.
Similarly, if $... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $-\mathbf A$ be the [[Definition:Negative Matrix|negative]] of $\mathbf A$.
If $\mathbf A$ is an [[Definition:Upper Triangular Matrix|upper triangular matrix]], then so is $-\mathbf A$... | From the definition of [[Definition:Negative Matrix|negative matrix]], we have:
:$\forall i, j \in \closedint 1 n: \sqbrk {-a}_{i j} = -a_{i j}$
If $\mathbf A$ is an [[Definition:Upper Triangular Matrix|upper triangular matrix]], we have:
:$\forall i > j: a_{i j} = 0$
Hence:
:$\forall i > j: \sqbrk {-a}_{i j} = -a_... | Negative of Triangular Matrix | https://proofwiki.org/wiki/Negative_of_Triangular_Matrix | https://proofwiki.org/wiki/Negative_of_Triangular_Matrix | [
"Negative Matrices",
"Triangular Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Negative Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix"
] | [
"Definition:Negative Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Category:Negative Matrices",
"Category:Triangular... |
proofwiki-3138 | Triangular Matrices forms Subring of Square Matrices | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map {\MM_R} n$ be the order $n$ square matrix space over a ring $R$.
Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$.
Let $\map {U_R} n$ be the set of upper triangular matrices of order $n$ over $R$.
Then $\ma... | From Negative of Triangular Matrix, if $\mathbf B \in \map {U_R} n$ then $-\mathbf B \in \map {U_R} n$.
Then from Sum of Triangular Matrices, if $\mathbf A, -\mathbf B \in \map {U_R} n$ then $\mathbf A + \paren {-\mathbf B} \in \map {U_R} n$.
From Product of Triangular Matrices, if $\mathbf A, \mathbf B \in \map {U_R} ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map {\MM_R} n$ be the [[Definition:Matrix Space|order $n$ square matrix space]] over a [[Definition:Ring (Abstract Algebra)|ring]] $R$.
Let $\struct {\map {\MM_R} n, +, \times}$ denote the [[Definition:Ring of Square ... | From [[Negative of Triangular Matrix]], if $\mathbf B \in \map {U_R} n$ then $-\mathbf B \in \map {U_R} n$.
Then from [[Sum of Triangular Matrices]], if $\mathbf A, -\mathbf B \in \map {U_R} n$ then $\mathbf A + \paren {-\mathbf B} \in \map {U_R} n$.
From [[Product of Triangular Matrices]], if $\mathbf A, \mathbf B \... | Triangular Matrices forms Subring of Square Matrices | https://proofwiki.org/wiki/Triangular_Matrices_forms_Subring_of_Square_Matrices | https://proofwiki.org/wiki/Triangular_Matrices_forms_Subring_of_Square_Matrices | [
"Triangular Matrices",
"Rings of Square Matrices",
"Subrings"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Matrix Space",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring of Square Matrices",
"Definition:Set",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Subring",
"Definition:Set",
... | [
"Negative of Triangular Matrix",
"Sum of Triangular Matrices",
"Product of Triangular Matrices",
"Subring Test"
] |
proofwiki-3139 | Natural Numbers form Subsemiring of Integers | The semiring of natural numbers $\struct {\N, +, \times}$ forms a subsemiring of the ring of integers $\struct {\Z, +, \times}$. | We have that Natural Numbers form Commutative Semiring.
From Natural Numbers are Non-Negative Integers we have that $\N$ is a subset of $\Z$.
Hence the result, from the definition of subsemiring.
{{qed}}
Category:Natural Numbers
Category:Integers
0io25lnk5ngnlb90vw3fp3kag33fsht | The [[Definition:Semiring of Natural Numbers|semiring of natural numbers]] $\struct {\N, +, \times}$ forms a [[Definition:Subsemiring|subsemiring]] of the [[Definition:Ring of Integers|ring of integers]] $\struct {\Z, +, \times}$. | We have that [[Natural Numbers form Commutative Semiring]].
From [[Natural Numbers are Non-Negative Integers]] we have that $\N$ is a [[Definition:Subset|subset]] of $\Z$.
Hence the result, from the definition of [[Definition:Subsemiring|subsemiring]].
{{qed}}
[[Category:Natural Numbers]]
[[Category:Integers]]
0io25... | Natural Numbers form Subsemiring of Integers | https://proofwiki.org/wiki/Natural_Numbers_form_Subsemiring_of_Integers | https://proofwiki.org/wiki/Natural_Numbers_form_Subsemiring_of_Integers | [
"Natural Numbers",
"Integers"
] | [
"Definition:Semiring of Natural Numbers",
"Definition:Subsemiring",
"Definition:Ring of Integers"
] | [
"Natural Numbers form Commutative Semiring",
"Natural Numbers are Non-Negative Integers",
"Definition:Subset",
"Definition:Subsemiring",
"Category:Natural Numbers",
"Category:Integers"
] |
proofwiki-3140 | Integers form Subdomain of Rationals | The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers. | The rational numbers are defined as the quotient field of the integers.
From its method of construction, it follows that the integers $\Z$ are a subset of the rational numbers $\Q$.
Hence the result, from the definition of subdomain.
{{qed}} | The [[Integers form Integral Domain|integral domain of integers]] $\left({\Z, +, \times}\right)$ forms a [[Definition:Subdomain|subdomain]] of the [[Definition:Field of Rational Numbers|field of rational numbers]]. | The rational numbers are defined as the [[Definition:Rational Number|quotient field of the integers]].
From its method of construction, it follows that the [[Definition:Integer|integers]] $\Z$ are a [[Definition:Subset|subset]] of the [[Definition:Rational Number|rational numbers]] $\Q$.
Hence the result, from the de... | Integers form Subdomain of Rationals | https://proofwiki.org/wiki/Integers_form_Subdomain_of_Rationals | https://proofwiki.org/wiki/Integers_form_Subdomain_of_Rationals | [
"Integers",
"Rational Numbers"
] | [
"Integers form Integral Domain",
"Definition:Subdomain",
"Definition:Field of Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Integer",
"Definition:Subset",
"Definition:Rational Number",
"Definition:Subdomain"
] |
proofwiki-3141 | Real Numbers form Subfield of Complex Numbers | The field of real numbers $\struct {\R, +, \times}$ forms a subfield of the field of complex numbers $\struct {\C, +, \times}$. | From Additive Group of Reals is Normal Subgroup of Complex, $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$.
From Multiplicative Group of Reals is Normal Subgroup of Complex, $\struct {\R, \times}$ is a subgroup of $\struct {\C, \times}$.
The result follows from the Subfield Test via the One-Step Subgroup Test and... | The [[Definition:Field of Real Numbers|field of real numbers]] $\struct {\R, +, \times}$ forms a [[Definition:Subfield|subfield]] of the [[Definition:Field of Complex Numbers|field of complex numbers]] $\struct {\C, +, \times}$. | From [[Additive Group of Reals is Normal Subgroup of Complex]], $\struct {\R, +}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\C, +}$.
From [[Multiplicative Group of Reals is Normal Subgroup of Complex]], $\struct {\R, \times}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\C, \times}$.
The result follows... | Real Numbers form Subfield of Complex Numbers | https://proofwiki.org/wiki/Real_Numbers_form_Subfield_of_Complex_Numbers | https://proofwiki.org/wiki/Real_Numbers_form_Subfield_of_Complex_Numbers | [
"Examples of Subfields",
"Real Numbers",
"Complex Numbers"
] | [
"Definition:Field of Real Numbers",
"Definition:Subfield",
"Definition:Field of Complex Numbers"
] | [
"Additive Group of Reals is Normal Subgroup of Complex",
"Definition:Subgroup",
"Multiplicative Group of Reals is Normal Subgroup of Complex",
"Definition:Subgroup",
"Subfield Test",
"One-Step Subgroup Test",
"Two-Step Subgroup Test"
] |
proofwiki-3142 | Sum of All Ring Products is Additive Subgroup | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +}$ and $\struct {T, +}$ be additive subgroups of $\struct {R, +, \circ}$.
Let $S + T$ be defined as subset product.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \closedint 1 n}$
Then both $S + T$ and... | As $\struct {R, +}$ is abelian (from the definition of a ring), we have:
:$S + T = T + S$
from Subset Product of Commutative is Commutative.
So from Subset Product of Subgroups it follows that $S + T$ is an additive subgroup of $\struct {R, +, \circ}$.
Let $x, y \in S T$.
We have that $\struct {S T, +}$ is closed.
So $... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {S, +}$ and $\struct {T, +}$ be [[Definition:Additive Subgroup|additive subgroups]] of $\struct {R, +, \circ}$.
Let $S + T$ be defined as [[Definition:Subset Product|subset product]].
Let $S T$ be defined as:
:$\ds S T = \set ... | As $\struct {R, +}$ is [[Definition:Abelian Group|abelian]] (from the definition of a [[Definition:Ring (Abstract Algebra)|ring]]), we have:
:$S + T = T + S$
from [[Subset Product of Commutative is Commutative]].
So from [[Subset Product of Subgroups]] it follows that $S + T$ is an [[Definition:Additive Subgroup|addit... | Sum of All Ring Products is Additive Subgroup | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Additive_Subgroup | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Additive_Subgroup | [
"Ring Theory",
"Subset Products"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Additive Subgroup",
"Definition:Subset Product",
"Definition:Additive Subgroup"
] | [
"Definition:Abelian Group",
"Definition:Ring (Abstract Algebra)",
"Subset Product within Commutative Structure is Commutative",
"Subset Product of Subgroups",
"Definition:Additive Subgroup",
"Sum of All Ring Products is Closed under Addition",
"Two-Step Subgroup Test",
"Definition:Additive Subgroup"
] |
proofwiki-3143 | Sum of All Ring Products is Closed under Addition | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +}$ and $\struct {T, +}$ be additive subgroups of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \closedint 1 n}$
Then $\struct {S T, +}$ is a closed subset of $\struct {R, +}$. | Let $x_1, x_2 \in S T$.
Then:
:$\ds x_1 = \sum_{i \mathop = 1}^j s_i \circ t_i, x_2 = \sum_{i \mathop = 1}^k s_i \circ t_i$
for some $s_i, t_i, j, k$, etc.
By renaming the indices, we can express $x_2$ as:
:$\ds x_2 = \sum_{i \mathop = j + 1}^{j + k} s_i \circ t_i$
and hence:
:$\ds x_1 + x_2 = \sum_{i \mathop = 1}^j s_... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {S, +}$ and $\struct {T, +}$ be [[Definition:Additive Subgroup|additive subgroups]] of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \close... | Let $x_1, x_2 \in S T$.
Then:
:$\ds x_1 = \sum_{i \mathop = 1}^j s_i \circ t_i, x_2 = \sum_{i \mathop = 1}^k s_i \circ t_i$
for some $s_i, t_i, j, k$, etc.
By renaming the indices, we can express $x_2$ as:
:$\ds x_2 = \sum_{i \mathop = j + 1}^{j + k} s_i \circ t_i$
and hence:
:$\ds x_1 + x_2 = \sum_{i \mathop = 1}^j... | Sum of All Ring Products is Closed under Addition | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Closed_under_Addition | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Closed_under_Addition | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Additive Subgroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-3144 | Sum of All Ring Products is Associative | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +}, \struct {T, +}, \struct {U, +}$ be additive subgroups of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \closedint 1 n}$
Then:
:$\paren {S T} U = S \paren {T U}$ | We have by definition that $S T$ is made up of all finite sums of elements of the form $s \circ t$ where $s \in S, t \in T$.
From Sum of All Ring Products is Closed under Addition, this set is closed under ring addition.
Therefore, so are $\paren {S T} U$ and $S \paren {T U}$.
Let $z \in \paren {S T} U$.
Then $z$ is a ... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {S, +}, \struct {T, +}, \struct {U, +}$ be [[Definition:Additive Subgroup|additive subgroups]] of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i... | We have by definition that $S T$ is made up of all finite sums of elements of the form $s \circ t$ where $s \in S, t \in T$.
From [[Sum of All Ring Products is Closed under Addition]], this set is [[Definition:Closure (Abstract Algebra)|closed]] under [[Definition:Ring Addition|ring addition]].
Therefore, so are $\pa... | Sum of All Ring Products is Associative | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Associative | https://proofwiki.org/wiki/Sum_of_All_Ring_Products_is_Associative | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Additive Subgroup"
] | [
"Sum of All Ring Products is Closed under Addition",
"Definition:Closure (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)",
"Definition:Associative Operation",
"Sum of All Ring Products is Closed under Addition",
"Definition:Set Equality/Definition ... |
proofwiki-3145 | Sum of Ring Products is Subring of Commutative Ring | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $\struct {S, +, \circ}$ and $\struct {T, +, \circ}$ be subrings of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \closedint 1 n}$
Then $S T$ is a subring of $\struct {R, +, \cir... | From Sum of All Ring Products is Additive Subgroup we have that $\struct {S T, +}$ is an additive subgroup of $R$.
Let $x_1, x_2 \in S T$.
Then:
:$\ds x_1 = \sum_{i \mathop = 1}^m s_i \circ t_i, x_2 = \sum_{i \mathop = 1}^n s_j \circ t_j$
for some $s_i, t_i, s_j, t_j, m, n$, etc.
Then:
{{begin-eqn}}
{{eqn | l = x_1 \ci... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {S, +, \circ}$ and $\struct {T, +, \circ}$ be [[Definition:Subring|subrings]] of $\struct {R, +, \circ}$.
Let $S T$ be defined as:
:$\ds S T = \set {\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \closed... | From [[Sum of All Ring Products is Additive Subgroup]] we have that $\struct {S T, +}$ is an [[Definition:Additive Subgroup|additive subgroup]] of $R$.
Let $x_1, x_2 \in S T$.
Then:
:$\ds x_1 = \sum_{i \mathop = 1}^m s_i \circ t_i, x_2 = \sum_{i \mathop = 1}^n s_j \circ t_j$
for some $s_i, t_i, s_j, t_j, m, n$, etc.... | Sum of Ring Products is Subring of Commutative Ring | https://proofwiki.org/wiki/Sum_of_Ring_Products_is_Subring_of_Commutative_Ring | https://proofwiki.org/wiki/Sum_of_Ring_Products_is_Subring_of_Commutative_Ring | [
"Commutative Rings",
"Subrings"
] | [
"Definition:Commutative Ring",
"Definition:Subring",
"Definition:Subring"
] | [
"Sum of All Ring Products is Additive Subgroup",
"Definition:Additive Subgroup",
"Definition:Commutative/Operation",
"Subring Test"
] |
proofwiki-3146 | Complex Numbers form Subfield of Quaternions | The field of complex numbers $\struct {\C, +, \times}$ is isomorphic to the subfields of the quaternions $\struct {\mathbb H, +, \times}$ whose underlying subsets are:
:$(1): \quad \mathbb H_\mathbf i = \set {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: c = d = 0}$
:$(2): \quad \mathbb H_\mathbf... | Let $\phi_i: \mathbb H_\mathbf i \to \C$ be defined as:
:$\forall \mathbf x = \mathbf a \mathbf 1 + b \mathbf i \in \mathbb H_\mathbf i: \map {\phi_i} {\mathbf x} = a + b i$
where in this context $i$ is the imaginary unit.
Similarly we can define $\phi_j$ and $\phi_k$:
:$\forall \mathbf x = \mathbf a \mathbf 1 + c \ma... | The [[Definition:Field of Complex Numbers|field of complex numbers]] $\struct {\C, +, \times}$ is [[Definition:Field Isomorphism|isomorphic]] to the [[Definition:Subfield|subfields]] of the [[Definition:Quaternion|quaternions]] $\struct {\mathbb H, +, \times}$ whose [[Definition:Underlying Set of Structure|underlying s... | Let $\phi_i: \mathbb H_\mathbf i \to \C$ be defined as:
:$\forall \mathbf x = \mathbf a \mathbf 1 + b \mathbf i \in \mathbb H_\mathbf i: \map {\phi_i} {\mathbf x} = a + b i$
where in this context $i$ is the [[Definition:Imaginary Unit|imaginary unit]].
Similarly we can define $\phi_j$ and $\phi_k$:
:$\forall \mathbf... | Complex Numbers form Subfield of Quaternions | https://proofwiki.org/wiki/Complex_Numbers_form_Subfield_of_Quaternions | https://proofwiki.org/wiki/Complex_Numbers_form_Subfield_of_Quaternions | [
"Complex Numbers",
"Quaternions",
"Subfields"
] | [
"Definition:Field of Complex Numbers",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Subfield",
"Definition:Quaternion",
"Definition:Underlying Set/Abstract Algebra"
] | [
"Definition:Complex Number/Imaginary Unit"
] |
proofwiki-3147 | Ring Homomorphism whose Kernel contains Ideal | Let $R$ be a ring.
Let $J$ be an ideal of $R$.
Let $\nu: R \to R / J$ be the quotient epimorphism.
Let $\phi: R \to S$ be a ring homomorphism such that:
:$J \subseteq \map \ker \phi$
where $\map \ker \phi$ is the kernel of $\phi$.
Then there exists a unique ring homomorphism $\psi: R / J \to S$ such that:
:$\phi = \psi... | Suppose $\phi = \psi \circ \nu$.
Let $J + x$ be an arbitrary element of $R / J$.
Then:
:$(1) \quad \map \psi {J + x} = \psi \circ \map \nu x = \map \phi x$
So there is only one possible way to define $\psi$.
Now suppose $J + x = J + x'$.
Then $x + \paren {-x'} \in J$.
So $x + \paren {-x'} \in \map \ker \phi$ as $J \sub... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $\nu: R \to R / J$ be the [[Definition:Quotient Ring Epimorphism|quotient epimorphism]].
Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]] such that:
:$J \subseteq \map \k... | Suppose $\phi = \psi \circ \nu$.
Let $J + x$ be an arbitrary element of $R / J$.
Then:
:$(1) \quad \map \psi {J + x} = \psi \circ \map \nu x = \map \phi x$
So there is only one possible way to define $\psi$.
Now suppose $J + x = J + x'$.
Then $x + \paren {-x'} \in J$.
So $x + \paren {-x'} \in \map \ker \phi$ as $J... | Ring Homomorphism whose Kernel contains Ideal | https://proofwiki.org/wiki/Ring_Homomorphism_whose_Kernel_contains_Ideal | https://proofwiki.org/wiki/Ring_Homomorphism_whose_Kernel_contains_Ideal | [
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Quotient Epimorphism/Ring",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Composition of Mappings"
] | [
"Definition:Well-Defined/Mapping",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring Homomorphism"
] |
proofwiki-3148 | Decay Equation | The first order ordinary differential equation:
:$\dfrac {\d y} {\d x} = k \paren {y_a - y}$
where $k \in \R: k > 0$
has the general solution:
:$y = y_a + C e^{-k x}$
where $C$ is an arbitrary constant.
If $y = y_0$ at $x = 0$, then:
:$y = y_a + \paren {y_0 - y_a} e^{-k x}$
This differential equation is known as the ''... | {{begin-eqn}}
{{eqn | l = \frac {\d y} {\d x}
| r = -k \paren {y - y_a}
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d y} {y - y_a}
| r = -\int k \rd x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = \map \ln {y - y_a}
| r = -k x + C_1
... | The [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$\dfrac {\d y} {\d x} = k \paren {y_a - y}$
where $k \in \R: k > 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = y_a + C e^{-k x}$
where $C$ is an [[Definition:Arbitra... | {{begin-eqn}}
{{eqn | l = \frac {\d y} {\d x}
| r = -k \paren {y - y_a}
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d y} {y - y_a}
| r = -\int k \rd x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {y - y_a}
| r = -k x + C_1
... | Decay Equation | https://proofwiki.org/wiki/Decay_Equation | https://proofwiki.org/wiki/Decay_Equation | [
"Decay Equation",
"Examples of Solutions to Separable Differential Equation",
"Examples of Linear First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Solution to Separable Differential Equation",
"Primitive of Reciprocal",
"Derivatives of Function of a x + b",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Initial Condition"
] |
proofwiki-3149 | Relation on Empty Set is Equivalence | Let $S = \O$, that is, the empty set.
Let $\RR \subseteq S \times S$ be a relation on $S$.
Then $\RR$ is the null relation and is an equivalence relation. | As $S = \O$, we have from Cartesian Product is Empty iff Factor is Empty that $S \times S = \O$.
Then it follows that $\RR \subseteq S \times S = \O$. | Let $S = \O$, that is, the [[Definition:Empty Set|empty set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] on $S$.
Then $\RR$ is the [[Definition:Null Relation|null relation]] and is an [[Definition:Equivalence Relation|equivalence relation]]. | As $S = \O$, we have from [[Cartesian Product is Empty iff Factor is Empty]] that $S \times S = \O$.
Then it follows that $\RR \subseteq S \times S = \O$. | Relation on Empty Set is Equivalence | https://proofwiki.org/wiki/Relation_on_Empty_Set_is_Equivalence | https://proofwiki.org/wiki/Relation_on_Empty_Set_is_Equivalence | [
"Examples of Equivalence Relations",
"Empty Set"
] | [
"Definition:Empty Set",
"Definition:Relation",
"Definition:Null Relation",
"Definition:Equivalence Relation"
] | [
"Cartesian Product is Empty iff Factor is Empty"
] |
proofwiki-3150 | Fourth Isomorphism Theorem | Let $\phi: R \to S$ be a ring homomorphism.
Let $K = \map \ker \phi$ be the kernel of $\phi$.
Let $\mathbb K$ be the set of all subrings of $R$ which contain $K$ as a subset.
Let $\mathbb S$ be the set of all subrings of $\Img \phi$.
Let $\phi^\to: \powerset R \to \powerset S$ be the direct image mapping of $\phi$.
The... | === Proof of Preservation of Subsets ===
From Subset Maps to Subset, we have:
:$(a) \quad \forall X, X' \in \mathbb K: X \subseteq X' \implies \map {\phi^\to} X \subseteq \map {\phi^\to} {X'}$
:$(b) \quad \forall Y, Y' \in \mathbb S: Y \subseteq Y' \implies \map {\paren {\phi^\to}^{-1} } Y \subseteq \map {\paren {\phi^... | Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $K = \map \ker \phi$ be the [[Definition:Kernel of Ring Homomorphism|kernel]] of $\phi$.
Let $\mathbb K$ be the [[Definition:Set|set]] of all [[Definition:Subring|subrings]] of $R$ which contain $K$ as a [[Definition:Subset|subset]].
L... | === Proof of Preservation of Subsets ===
From [[Subset Maps to Subset]], we have:
:$(a) \quad \forall X, X' \in \mathbb K: X \subseteq X' \implies \map {\phi^\to} X \subseteq \map {\phi^\to} {X'}$
:$(b) \quad \forall Y, Y' \in \mathbb S: Y \subseteq Y' \implies \map {\paren {\phi^\to}^{-1} } Y \subseteq \map {\paren ... | Fourth Isomorphism Theorem | https://proofwiki.org/wiki/Fourth_Isomorphism_Theorem | https://proofwiki.org/wiki/Fourth_Isomorphism_Theorem | [
"Quotient Rings",
"Named Theorems",
"Isomorphisms (Abstract Algebra)",
"Isomorphism Theorems"
] | [
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Set",
"Definition:Subring",
"Definition:Subset",
"Definition:Set",
"Definition:Subring",
"Definition:Direct Image Mapping/Mapping",
"Definition:Restriction/Mapping",
"Definition:Bijection",
"Definition:Inverse ... | [
"Image of Subset under Mapping is Subset of Image",
"Definition:Inverse Mapping",
"Definition:Subset"
] |
proofwiki-3151 | Increasing Union of Subrings is Subring | Let $R$ be a ring.
Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ be subrings of $R$.
Then the increasing union $S$:
:$\ds S = \bigcup_{i \mathop \in \N} S_i$
is a subring of $R$. | Let $\ds S = \bigcup_{i \mathop \in \N} S_i$.
Clearly $0_R \in S$.
Let $a, b \in S$.
Then $\exists i, j \in \N: a \in S_i, b \in S_j$.
From the construction, we have that either of $S_i$ and $S_j$ contains the other.
Let $l = \max \set {i, j}$ so $a, b \in S_l$.
Then $a + \paren {-b} \in S_l$ and $a b \in S_l$, as $S_l... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ be [[Definition:Subring|subrings]] of $R$.
Then the [[Definition:Increasing Union|increasing union]] $S$:
:$\ds S = \bigcup_{i \mathop \in \N} S_i$
is a [[Definition:Subring... | Let $\ds S = \bigcup_{i \mathop \in \N} S_i$.
Clearly $0_R \in S$.
Let $a, b \in S$.
Then $\exists i, j \in \N: a \in S_i, b \in S_j$.
From the construction, we have that either of $S_i$ and $S_j$ contains the other.
Let $l = \max \set {i, j}$ so $a, b \in S_l$.
Then $a + \paren {-b} \in S_l$ and $a b \in S_l$, a... | Increasing Union of Subrings is Subring | https://proofwiki.org/wiki/Increasing_Union_of_Subrings_is_Subring | https://proofwiki.org/wiki/Increasing_Union_of_Subrings_is_Subring | [
"Set Union",
"Subrings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subring",
"Definition:Increasing Union",
"Definition:Subring"
] | [
"Definition:Subring",
"Subring Test",
"Definition:Subring"
] |
proofwiki-3152 | Subrings of Integers are Sets of Integer Multiples | Let $\struct {\Z, +, \times}$ be the integral domain of integers.
The subrings of $\struct {\Z, +, \times}$ are the rings of integer multiples:
:$\struct {n \Z, +, \times}$
where $n \in \Z: n \ge 0$.
There are no other subrings of $\struct {\Z, +, \times}$ but these. | From Integer Multiples form Commutative Ring, it is clear that $\struct {n \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$ when $n \ge 1$.
We also note that when $n = 0$, we have:
:$\struct {n \Z, +, \times} = \struct {0, +, \times}$
which is the null ring.
When $n = 1$, we have:
:$\struct {n \Z, +, \times} =... | Let $\struct {\Z, +, \times}$ be the [[Integers form Integral Domain|integral domain of integers]].
The [[Definition:Subring|subrings]] of $\struct {\Z, +, \times}$ are the [[Integer Multiples form Commutative Ring|rings of integer multiples]]:
:$\struct {n \Z, +, \times}$
where $n \in \Z: n \ge 0$.
There are no ot... | From [[Integer Multiples form Commutative Ring]], it is clear that $\struct {n \Z, +, \times}$ is a [[Definition:Subring|subring]] of $\struct {\Z, +, \times}$ when $n \ge 1$.
We also note that when $n = 0$, we have:
:$\struct {n \Z, +, \times} = \struct {0, +, \times}$
which is the [[Definition:Null Ring|null ring]].... | Subrings of Integers are Sets of Integer Multiples | https://proofwiki.org/wiki/Subrings_of_Integers_are_Sets_of_Integer_Multiples | https://proofwiki.org/wiki/Subrings_of_Integers_are_Sets_of_Integer_Multiples | [
"Subrings of Integers are Sets of Integer Multiples",
"Subrings",
"Integers"
] | [
"Integers form Integral Domain",
"Definition:Subring",
"Integer Multiples form Commutative Ring",
"Definition:Subring"
] | [
"Integer Multiples form Commutative Ring",
"Definition:Subring",
"Definition:Null Ring",
"Null Ring and Ring Itself are Subrings",
"Definition:Subring",
"Subgroups of Additive Group of Integers",
"Definition:Additive Subgroup",
"Definition:Subring",
"Definition:Additive Group of Ring"
] |
proofwiki-3153 | Subring of Integers is Ideal | Every subring of $\struct {\Z, +, \times}$ is an ideal of the ring $\struct {\Z, +, \times}$. | Follows directly from:
:Subrings of Integers are Sets of Integer Multiples
and:
:Subgroup of Integers is Ideal.
{{qed}} | Every [[Definition:Subring|subring]] of $\struct {\Z, +, \times}$ is an [[Definition:Ideal of Ring|ideal]] of the [[Definition:Ring (Abstract Algebra)|ring]] $\struct {\Z, +, \times}$. | Follows directly from:
:[[Subrings of Integers are Sets of Integer Multiples]]
and:
:[[Subgroup of Integers is Ideal]].
{{qed}} | Subring of Integers is Ideal | https://proofwiki.org/wiki/Subring_of_Integers_is_Ideal | https://proofwiki.org/wiki/Subring_of_Integers_is_Ideal | [
"Subrings",
"Integers",
"Ideal Theory"
] | [
"Definition:Subring",
"Definition:Ideal of Ring",
"Definition:Ring (Abstract Algebra)"
] | [
"Subrings of Integers are Sets of Integer Multiples",
"Subgroup of Integers is Ideal"
] |
proofwiki-3154 | Polynomial Ring of Sequences is Ring | Let $R$ be a ring.
Let $P \sqbrk R$ be the polynomial ring over sequences in $R$.
Then $P \sqbrk R$ is itself a ring. | We have by definition of polynomial ring over sequences in $R$ that:
:$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$
where each $r_i \in R$, and all but a finite number of terms is zero. | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $P \sqbrk R$ be the [[Definition:Polynomial Ring over Sequences|polynomial ring over sequences in $R$]].
Then $P \sqbrk R$ is itself a [[Definition:Ring (Abstract Algebra)|ring]]. | We have by definition of [[Definition:Polynomial Ring over Sequences|polynomial ring over sequences in $R$]] that:
:$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$
where each $r_i \in R$, and all but a [[Definition:Finite Set|finite number]] of [[Definition:Term of Sequence|terms]] is [[Definition:Ring Zero|z... | Polynomial Ring of Sequences is Ring | https://proofwiki.org/wiki/Polynomial_Ring_of_Sequences_is_Ring | https://proofwiki.org/wiki/Polynomial_Ring_of_Sequences_is_Ring | [
"Polynomial Theory",
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Polynomial Ring/Sequences",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Polynomial Ring/Sequences",
"Definition:Finite Set",
"Definition:Term of Sequence",
"Definition:Ring Zero",
"Definition:Ring Zero"
] |
proofwiki-3155 | Definition of Polynomial from Polynomial Ring over Sequences | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $\struct {P \sqbrk R, \oplus, \odot}$ be the polynomial ring over the set of all sequences in $R$:
:$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$
where the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as:
{{:Definition:Operations on Polynomi... | Let $P \sqbrk R$ be the polynomial ring over $R$.
Consider the injection $\phi: R \to P \sqbrk R$ defined as:
:$\forall r \in R: \map \phi r = \sequence {r, 0, 0, \ldots}$
It is easily checked that $\phi$ is a ring monomorphism.
So the set $\set {\sequence {r, 0, 0, \ldots}: r \in R}$ is a subring of $P \sqbrk R$ which... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $\struct {P \sqbrk R, \oplus, \odot}$ be the [[Definition:Polynomial Ring over Sequences|polynomial ring]] over the [[Definition:Set|set]] of all [[Definition:Sequence|sequences in $R$]]:
:$P \sqbrk R = \set {\sequence {r_0, r_1, r_2... | Let $P \sqbrk R$ be the [[Definition:Polynomial Ring over Sequences|polynomial ring over $R$]].
Consider the [[Definition:Injection|injection]] $\phi: R \to P \sqbrk R$ defined as:
:$\forall r \in R: \map \phi r = \sequence {r, 0, 0, \ldots}$
It is easily checked that $\phi$ is a [[Definition:Ring Monomorphism|ring m... | Definition of Polynomial from Polynomial Ring over Sequences | https://proofwiki.org/wiki/Definition_of_Polynomial_from_Polynomial_Ring_over_Sequences | https://proofwiki.org/wiki/Definition_of_Polynomial_from_Polynomial_Ring_over_Sequences | [
"Ring Theory",
"Polynomial Theory"
] | [
"Definition:Ring with Unity",
"Definition:Polynomial Ring/Sequences",
"Definition:Set",
"Definition:Sequence",
"Definition:Operations on Polynomial Ring of Sequences",
"Definition:Polynomial Ring",
"Definition:Polynomial over Ring as Function on Free Monoid on Set",
"Definition:Polynomial Ring/Indeter... | [
"Definition:Polynomial Ring/Sequences",
"Definition:Injection",
"Definition:Ring Monomorphism",
"Definition:Subring",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Sequence",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Polynomial Ring/Sequences",
"Definition... |
proofwiki-3156 | Integers under Multiplication form Countably Infinite Semigroup | The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite semigroup. | From Integers under Multiplication form Semigroup, $\struct {\Z, \times}$ is a countably infinite semigroup.
Then we have that the Integers are Countably Infinite.
The criteria for $\struct {\Z, \times}$ to be a countably infinite semigroup are seen to be satisfied.
{{Qed}}
Category:Integer Multiplication
Category:Exa... | The [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Multiplication|multiplication]] $\struct {\Z, \times}$ is a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Semigroup|semigroup]]. | From [[Integers under Multiplication form Semigroup]], $\struct {\Z, \times}$ is a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Semigroup|semigroup]].
Then we have that the [[Integers are Countably Infinite]].
The criteria for $\struct {\Z, \times}$ to be a [[Definition:Countably Infinite Set... | Integers under Multiplication form Countably Infinite Semigroup | https://proofwiki.org/wiki/Integers_under_Multiplication_form_Countably_Infinite_Semigroup | https://proofwiki.org/wiki/Integers_under_Multiplication_form_Countably_Infinite_Semigroup | [
"Integer Multiplication",
"Examples of Semigroups"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Countably Infinite/Set",
"Definition:Semigroup"
] | [
"Integers under Multiplication form Semigroup",
"Definition:Countably Infinite/Set",
"Definition:Semigroup",
"Integers are Countably Infinite",
"Definition:Countably Infinite/Set",
"Definition:Semigroup",
"Category:Integer Multiplication",
"Category:Examples of Semigroups"
] |
proofwiki-3157 | Integer Addition is Closed | The set of integers is closed under addition:
:$\forall a, b \in \Z: a + b \in \Z$ | Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the formal definition of integers.
That is, $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.
$\boxminus$ is the congruence relation defined on $\N \times \N$ by:
:$\tuple... | The [[Definition:Set|set]] of [[Definition:Integer|integers]] is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Integer Addition|addition]]:
:$\forall a, b \in \Z: a + b \in \Z$ | Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the [[Definition:Integer/Formal Definition|formal definition of integers]].
That is, $\eqclass {\tuple {a, b} } \boxminus$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natu... | Integer Addition is Closed | https://proofwiki.org/wiki/Integer_Addition_is_Closed | https://proofwiki.org/wiki/Integer_Addition_is_Closed | [
"Integer Addition",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Integers"
] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Integer/Formal Definition/Notation",
"Definition:Addition/Integers",
"Definition:Addition/I... |
proofwiki-3158 | Integer Addition Identity is Zero | The identity of integer addition is $0$:
:$\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$ | Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers.
That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.
$\boxtimes$ is the congruence relation defined on $\N \times \N$ by:
:$\tuple... | The [[Definition:Identity Element|identity]] of [[Definition:Integer Addition|integer addition]] is $0$:
:$\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$ | Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the [[Definition:Integer/Formal Definition|formal definition of integers]].
That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natu... | Integer Addition Identity is Zero | https://proofwiki.org/wiki/Integer_Addition_Identity_is_Zero | https://proofwiki.org/wiki/Integer_Addition_Identity_is_Zero | [
"Integer Addition"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Addition/Integers"
] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Integer/Formal Definition/Notation",
"Construction of Inverse Completion",
"Definition:Natu... |
proofwiki-3159 | Inverse for Integer Addition | Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:
:$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$ | Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers.
That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.
$\boxtimes$ is the congruence relation defined on $\N \times \N$ by:
:$\tuple... | Each element $x$ of the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ has an [[Definition:Inverse Element|inverse element]] $-x$ under the operation of [[Definition:Integer Addition|integer addition]]:
:$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$ | Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the [[Definition:Integer/Formal Definition|formal definition of integers]].
That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natu... | Inverse for Integer Addition | https://proofwiki.org/wiki/Inverse_for_Integer_Addition | https://proofwiki.org/wiki/Inverse_for_Integer_Addition | [
"Integer Addition",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Addition/Integers"
] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Integer/Formal Definition/Notation",
"Construction of Inverse Completion/Invertible Elements ... |
proofwiki-3160 | Rational Numbers form Ring | The set of rational numbers $\Q$ forms a ring under addition and multiplication: $\struct {\Q, +, \times}$. | Recall that $\struct {\Q, +, \times}$ is a field.
As a field is also by definition a division ring, which is an example of a ring, the result follows.
{{qed}} | The [[Definition:Rational Number|set of rational numbers]] $\Q$ forms a [[Definition:Ring (Abstract Algebra)|ring]] under [[Definition:Rational Addition|addition]] and [[Definition:Rational Multiplication|multiplication]]: $\struct {\Q, +, \times}$. | Recall that [[Rational Numbers form Field|$\struct {\Q, +, \times}$ is a field]].
As a [[Definition:Field (Abstract Algebra)|field]] is also by definition a [[Definition:Division Ring|division ring]], which is an example of a [[Definition:Ring (Abstract Algebra)|ring]], the result follows.
{{qed}} | Rational Numbers form Ring | https://proofwiki.org/wiki/Rational_Numbers_form_Ring | https://proofwiki.org/wiki/Rational_Numbers_form_Ring | [
"Examples of Rings",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Ring (Abstract Algebra)",
"Definition:Addition/Rational Numbers",
"Definition:Multiplication/Rational Numbers"
] | [
"Rational Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Division Ring",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-3161 | Real Numbers form Ring | The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$. | From Real Numbers under Addition form Infinite Abelian Group, $\struct {\R, +}$ is an abelian group.
We also have that:
:Real Multiplication is Closed:
::$\forall x, y \in \R: x \times y \in \R$
:Real Multiplication is Associative:
::$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$
... | The [[Definition:Real Number|set of real numbers]] $\R$ forms a [[Definition:Ring (Abstract Algebra)|ring]] under [[Definition:Real Addition|addition]] and [[Definition:Real Multiplication|multiplication]]: $\struct {\R, +, \times}$. | From [[Real Numbers under Addition form Infinite Abelian Group]], $\struct {\R, +}$ is an [[Definition:Abelian Group|abelian group]].
We also have that:
:[[Real Multiplication is Closed]]:
::$\forall x, y \in \R: x \times y \in \R$
:[[Real Multiplication is Associative]]:
::$\forall x, y, z \in \R: x \times \paren {... | Real Numbers form Ring | https://proofwiki.org/wiki/Real_Numbers_form_Ring | https://proofwiki.org/wiki/Real_Numbers_form_Ring | [
"Examples of Rings",
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Ring (Abstract Algebra)",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers"
] | [
"Real Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Real Multiplication is Closed",
"Real Multiplication is Associative",
"Definition:Semigroup",
"Real Multiplication Distributes over Addition",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-3162 | Complex Numbers form Ring | The set of complex numbers $\C$ forms a ring under addition and multiplication: $\struct {\C, +, \times}$. | From Complex Numbers under Addition form Infinite Abelian Group, $\struct {\C, +}$ is an abelian group.
We also have that:
:Complex Multiplication is Closed:
::$\forall x, y \in \C: x \times y \in \C$
:Complex Multiplication is Associative:
::$\forall x, y, z \in \C: x \times \paren {y \times z} = \paren {x \times y} \... | The [[Definition:Complex Number|set of complex numbers]] $\C$ forms a [[Definition:Ring (Abstract Algebra)|ring]] under [[Definition:Complex Addition|addition]] and [[Definition:Complex Multiplication|multiplication]]: $\struct {\C, +, \times}$. | From [[Complex Numbers under Addition form Infinite Abelian Group]], $\struct {\C, +}$ is an [[Definition:Abelian Group|abelian group]].
We also have that:
:[[Complex Multiplication is Closed]]:
::$\forall x, y \in \C: x \times y \in \C$
:[[Complex Multiplication is Associative]]:
::$\forall x, y, z \in \C: x \times... | Complex Numbers form Ring | https://proofwiki.org/wiki/Complex_Numbers_form_Ring | https://proofwiki.org/wiki/Complex_Numbers_form_Ring | [
"Examples of Rings",
"Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Ring (Abstract Algebra)",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers"
] | [
"Complex Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Complex Multiplication is Closed",
"Complex Multiplication is Associative",
"Definition:Semigroup",
"Complex Multiplication Distributes over Addition",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-3163 | Complex Addition Identity is Zero | Let $\C$ be the set of complex numbers.
The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {x + i y} + \paren {0 + 0 i}
| r = \paren {x + 0} + i \paren {y + 0} = x + i y
}}
{{eqn | l = \paren {0 + 0 i} + \paren {x + i y}
| r = \paren {0 + x} + i \paren {0 + y} = x + i y
}}
{{end-eqn}}
{{qed}} | Let $\C$ be the set of [[Definition:Complex Number|complex numbers]].
The [[Definition:Identity Element|identity element]] of $\struct {\C, +}$ is the [[Definition:Complex Number|complex number]] $0 + 0 i$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {x + i y} + \paren {0 + 0 i}
| r = \paren {x + 0} + i \paren {y + 0} = x + i y
}}
{{eqn | l = \paren {0 + 0 i} + \paren {x + i y}
| r = \paren {0 + x} + i \paren {0 + y} = x + i y
}}
{{end-eqn}}
{{qed}} | Complex Addition Identity is Zero | https://proofwiki.org/wiki/Complex_Addition_Identity_is_Zero | https://proofwiki.org/wiki/Complex_Addition_Identity_is_Zero | [
"Complex Addition",
"Examples of Identity Elements"
] | [
"Definition:Complex Number",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Complex Number"
] | [] |
proofwiki-3164 | Rational Addition Identity is Zero | The identity of rational number addition is $0$:
:$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$ | From the definition, the field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
From Zero of Inverse Completion of Integral Domain, for any $k \in \Z^*$, the element $\dfrac {0_D} k$ of $\Q$ serves as the zero of $\struct {\Q, +, \time... | The [[Definition:Identity Element|identity]] of [[Definition:Rational Addition|rational number addition]] is $0$:
:$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$ | From the definition, the [[Definition:Field (Abstract Algebra)|field]] $\struct {\Q, +, \times}$ of [[Definition:Rational Number|rational numbers]] is the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers... | Rational Addition Identity is Zero | https://proofwiki.org/wiki/Rational_Addition_Identity_is_Zero | https://proofwiki.org/wiki/Rational_Addition_Identity_is_Zero | [
"Rational Addition"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Addition/Rational Numbers"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Zero of Inverse Completion of Integral Domain",
"Definition:Field Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Equal Elem... |
proofwiki-3165 | Inverse for Complex Addition | Let $z = x + i y \in \C$ be a complex number.
Let $-z = -x - i y \in \C$ be the negative of $z$.
Then $-z$ is the inverse element of $z$ under the operation of complex addition:
:$\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$ | From Complex Addition Identity is Zero, the identity element for $\struct {\C, +}$ is $0 + 0 i$.
Then:
{{begin-eqn}}
{{eqn | o =
| r = \paren {x + i y} + \paren {-x - i y}
| c =
}}
{{eqn | r = \paren {x - x} + i \paren {y - y}
| c =
}}
{{eqn | r = 0 + 0 i
| c =
}}
{{end-eqn}}
Similarly for ... | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]].
Let $-z = -x - i y \in \C$ be the [[Definition:Negative of Complex Number|negative]] of $z$.
Then $-z$ is the [[Definition:Inverse Element|inverse element]] of $z$ under the operation of [[Definition:Complex Addition|complex addition]]:
:$\fo... | From [[Complex Addition Identity is Zero]], the [[Definition:Identity Element|identity element]] for $\struct {\C, +}$ is $0 + 0 i$.
Then:
{{begin-eqn}}
{{eqn | o =
| r = \paren {x + i y} + \paren {-x - i y}
| c =
}}
{{eqn | r = \paren {x - x} + i \paren {y - y}
| c =
}}
{{eqn | r = 0 + 0 i
... | Inverse for Complex Addition | https://proofwiki.org/wiki/Inverse_for_Complex_Addition | https://proofwiki.org/wiki/Inverse_for_Complex_Addition | [
"Complex Addition",
"Examples of Inverse Elements"
] | [
"Definition:Complex Number",
"Definition:Negative/Complex Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Addition/Complex Numbers"
] | [
"Complex Addition Identity is Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-3166 | Equal Elements of Field of Quotients | Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.
Let $\struct {K, +, \circ}$ be the field of quotients of $\struct {D, +, \circ}$.
Let $x = \dfrac p q \in K$.
Then:
:$\forall k \in D^*: x = \dfrac {p \circ k} {q \circ k}$
where:
:$D^* := D \setminus \set {0_D}$
that is, $D$ with its zero removed. | We have that the field of quotients $\struct {K, +, \circ}$ of an integral domain is its inverse completion.
Thus we have:
:$\forall x_1, x_2 \in D, y_1, y_2 \in D^*: \dfrac {x_1} {y_1} = \dfrac {x_2} {y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$
So:
{{begin-eqn}}
{{eqn | l = \paren {p \circ k} \circ q
| r = p \circ ... | Let $\struct {D, +, \circ}$ be an [[Definition:Integral Domain|integral domain]] whose [[Definition:Ring Zero|zero]] is $0_D$.
Let $\struct {K, +, \circ}$ be the [[Definition:Field of Quotients|field of quotients]] of $\struct {D, +, \circ}$.
Let $x = \dfrac p q \in K$.
Then:
:$\forall k \in D^*: x = \dfrac {p \ci... | We have that the [[Definition:Field of Quotients|field of quotients]] $\struct {K, +, \circ}$ of an [[Definition:Integral Domain|integral domain]] is its [[Inverse Completion of Integral Domain Exists|inverse completion]].
Thus we have:
:$\forall x_1, x_2 \in D, y_1, y_2 \in D^*: \dfrac {x_1} {y_1} = \dfrac {x_2} {y_2... | Equal Elements of Field of Quotients | https://proofwiki.org/wiki/Equal_Elements_of_Field_of_Quotients | https://proofwiki.org/wiki/Equal_Elements_of_Field_of_Quotients | [
"Integral Domains",
"Inverse Completions",
"Fields of Quotients"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Field of Quotients",
"Definition:Ring Zero"
] | [
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Inverse Completion of Integral Domain Exists",
"Definition:Integral Domain",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Category:Integral Domains",
"Category:Inverse Completions",
"Category:Fields of Quotie... |
proofwiki-3167 | Rational Addition is Closed | The operation of addition on the set of rational numbers $\Q$ is well-defined and closed:
:$\forall x, y \in \Q: x + y \in \Q$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $+$ is well-defined and closed on $\Q$.
{{qed}} | The operation of [[Definition:Rational Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Well-Defined Operation|well-defined]] and [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in \Q: x + y \in \Q$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Addition is Closed | https://proofwiki.org/wiki/Rational_Addition_is_Closed | https://proofwiki.org/wiki/Rational_Addition_is_Closed | [
"Rational Addition",
"Algebraic Closure"
] | [
"Definition:Addition/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Well-Defined/Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:A Fortiori",
"Definition:Well-Defined/Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-3168 | Inverse for Rational Addition | Each element $x$ of the set of rational numbers $\Q$ has an inverse element $-x$ under the operation of rational number addition:
:$\forall x \in \Q: \exists -x \in \Q: x + \paren {-x} = 0 = \paren {-x} + x$ | Let $x = \dfrac a b$ where $b \ne 0$.
We take the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
From Existence of Field of Quotients, we have that the inverse of $\dfrac a b$ for $+$ is $\dfrac {-a} b$:
{{begin-eqn}}
{{eqn | l = \frac a b + \frac ... | Each element $x$ of the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ has an [[Definition:Inverse Element|inverse element]] $-x$ under the operation of [[Definition:Rational Addition|rational number addition]]:
:$\forall x \in \Q: \exists -x \in \Q: x + \paren {-x} = 0 = \paren {-x} + x... | Let $x = \dfrac a b$ where $b \ne 0$.
We take the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
From [[Existence of Field of Q... | Inverse for Rational Addition | https://proofwiki.org/wiki/Inverse_for_Rational_Addition | https://proofwiki.org/wiki/Inverse_for_Rational_Addition | [
"Rational Addition",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Addition/Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Existence of Field of Quotients",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Rational Addition Identity is Zero",
"Negative of Division Product",
"Definition:Inverse (Abstract A... |
proofwiki-3169 | Rational Multiplication is Closed | The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:
:$\forall x, y \in \Q: x \times y \in \Q$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore {{afortiori}} $\times$ is well-defined and closed on $\Q$.
{{qed}} | The operation of [[Definition:Rational Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Well-Defined Operation|well-defined]] and [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in \Q: x \times y \in \Q$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Multiplication is Closed/Proof 1 | https://proofwiki.org/wiki/Rational_Multiplication_is_Closed | https://proofwiki.org/wiki/Rational_Multiplication_is_Closed/Proof_1 | [
"Rational Multiplication",
"Algebraic Closure",
"Rational Multiplication is Closed"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Well-Defined/Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:Well-Defined/Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-3170 | Rational Multiplication is Closed | The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:
:$\forall x, y \in \Q: x \times y \in \Q$ | From the definition of rational numbers, there exists four integers $p$, $q$, $r$, $s$, where:
:$q \ne 0$
:$s \ne 0$
:$\dfrac p q = x$
:$\dfrac r s = y$
We have that:
:$p \times r \in \Z$
:$q \times s \in \Z$
Since $q \ne 0$ and $s \ne 0$, we have that:
:$q \times s \ne 0$
Therefore, by the definition of rational numbe... | The operation of [[Definition:Rational Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Well-Defined Operation|well-defined]] and [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in \Q: x \times y \in \Q$ | From the definition of [[Definition:Rational Number|rational numbers]], there exists four [[Definition:Integer|integers]] $p$, $q$, $r$, $s$, where:
:$q \ne 0$
:$s \ne 0$
:$\dfrac p q = x$
:$\dfrac r s = y$
We have that:
:$p \times r \in \Z$
:$q \times s \in \Z$
Since $q \ne 0$ and $s \ne 0$, we have that:
:$q \tim... | Rational Multiplication is Closed/Proof 2 | https://proofwiki.org/wiki/Rational_Multiplication_is_Closed | https://proofwiki.org/wiki/Rational_Multiplication_is_Closed/Proof_2 | [
"Rational Multiplication",
"Algebraic Closure",
"Rational Multiplication is Closed"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Well-Defined/Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Rational Number",
"Definition:Integer",
"Definition:Rational Number"
] |
proofwiki-3171 | Rational Multiplication Identity is One | The identity of rational number multiplication is $1$:
:$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$ | From the definition, the field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
From the properties of the quotient structure, elements of $\Q$ of the form $\dfrac p p$ where $p \ne 0$ act as the identity for multiplication.
From Equal... | The [[Definition:Identity Element|identity]] of [[Definition:Rational Multiplication|rational number multiplication]] is $1$:
:$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$ | From the definition, the [[Definition:Field (Abstract Algebra)|field]] $\struct {\Q, +, \times}$ of [[Definition:Rational Number|rational numbers]] is the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers... | Rational Multiplication Identity is One | https://proofwiki.org/wiki/Rational_Multiplication_Identity_is_One | https://proofwiki.org/wiki/Rational_Multiplication_Identity_is_One | [
"Rational Multiplication"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiplication/Rational Numbers"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Construction of Inverse Completion/Identity of Quotient Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multip... |
proofwiki-3172 | Inverse for Rational Multiplication | Each element $x$ of the set of non-zero rational numbers $\Q_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication:
:$\forall x \in \Q_{\ne 0}: \exists \dfrac 1 x \in \Q_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$ | From the definition, the field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
From Rational Multiplication Identity is One, the identity for $\struct {\Q, \times}$ is $1 = \dfrac 1 1 = \dfrac p p$ where $p \in \Z$ and $p \ne 0$.
From... | Each element $x$ of the [[Definition:Set|set]] of [[Definition:Rational Number|non-zero rational numbers]] $\Q_{\ne 0}$ has an [[Definition:Inverse Element|inverse element]] $\dfrac 1 x$ under the operation of [[Definition:Rational Multiplication|rational number multiplication]]:
:$\forall x \in \Q_{\ne 0}: \exists \df... | From the definition, the [[Definition:Field (Abstract Algebra)|field]] $\struct {\Q, +, \times}$ of [[Definition:Rational Number|rational numbers]] is the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers... | Inverse for Rational Multiplication | https://proofwiki.org/wiki/Inverse_for_Rational_Multiplication | https://proofwiki.org/wiki/Inverse_for_Rational_Multiplication | [
"Rational Multiplication",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Multiplication/Rational Numbers"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Rational Multiplication Identity is One",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Existence of Field of Quotients",
"Equal ... |
proofwiki-3173 | Real Addition Identity is Zero | The identity of real number addition is $0$:
:$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The [[Definition:Identity Element|identity]] of [[Definition:Real Addition|real number addition]] is $0$:
:$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Addition Identity is Zero | https://proofwiki.org/wiki/Real_Addition_Identity_is_Zero | https://proofwiki.org/wiki/Real_Addition_Identity_is_Zero | [
"Additive Group of Real Numbers",
"Real Addition",
"Zero"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Addition/Real Numbers"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Addition/Real Numbers",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Real Number"
] |
proofwiki-3174 | Inverse for Real Addition | Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:
:$\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$ | We have:
{{begin-eqn}}
{{eqn | l = \eqclass {\sequence {x_n} } {} + \paren {-\eqclass {\sequence {x_n} } {} }
| r = \eqclass {\sequence {x_n - x_n} } {}
| c =
}}
{{eqn | r = \eqclass {\sequence {0_n} } {}
| c =
}}
{{end-eqn}}
Similarly for $\paren {-\eqclass {\sequence {x_n} } {} } + \eqclass {\sequ... | Each element $x$ of the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ has an [[Definition:Inverse Element|inverse element]] $-x$ under the operation of [[Definition:Real Addition|real number addition]]:
:$\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$ | We have:
{{begin-eqn}}
{{eqn | l = \eqclass {\sequence {x_n} } {} + \paren {-\eqclass {\sequence {x_n} } {} }
| r = \eqclass {\sequence {x_n - x_n} } {}
| c =
}}
{{eqn | r = \eqclass {\sequence {0_n} } {}
| c =
}}
{{end-eqn}}
Similarly for $\paren {-\eqclass {\sequence {x_n} } {} } + \eqclass {\se... | Inverse for Real Addition | https://proofwiki.org/wiki/Inverse_for_Real_Addition | https://proofwiki.org/wiki/Inverse_for_Real_Addition | [
"Real Addition",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Addition/Real Numbers"
] | [
"Category:Real Addition",
"Category:Examples of Inverse Elements"
] |
proofwiki-3175 | Real Multiplication Identity is One | The identity element of real number multiplication is the real number $1$:
:$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$ | From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equiva... | The [[Definition:Identity Element|identity element]] of [[Definition:Real Multiplication|real number multiplication]] is the real number $1$:
:$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$ | From the definition, the [[Definition:Real Number|real numbers]] are the set of all [[Definition:Equivalence Class|equivalence classes]] $\eqclass {\sequence {x_n} } {}$ of [[Definition:Cauchy Sequence|Cauchy sequences]] of [[Definition:Rational Number|rational numbers]].
Let $x = \eqclass {\sequence {x_n} } {}, y = ... | Real Multiplication Identity is One | https://proofwiki.org/wiki/Real_Multiplication_Identity_is_One | https://proofwiki.org/wiki/Real_Multiplication_Identity_is_One | [
"Multiplicative Group of Real Numbers",
"Real Multiplication",
"1"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiplication/Real Numbers"
] | [
"Definition:Real Number",
"Definition:Equivalence Class",
"Definition:Cauchy Sequence",
"Definition:Rational Number",
"Definition:Equivalence Class",
"Definition:Multiplication/Real Numbers",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Real Number"
] |
proofwiki-3176 | Inverse for Real Multiplication | Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:
:$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$ | By the definition of real number:
:$\forall \epsilon \in \R_{>0}: \exists t \in \N: \forall i > t: \size {x_i - x} < \epsilon$
{{explain|In order for this line to make sense, it needs the definition (or at least the relevant part) extracted and posted here, so the context of the convergence condition is clear.}}
Let $\... | Each element $x$ of the [[Definition:Set|set]] of [[Definition:Real Number|non-zero real numbers]] $\R_{\ne 0}$ has an [[Definition:Inverse Element|inverse element]] $\dfrac 1 x$ under the operation of [[Definition:Real Multiplication|real number multiplication]]:
:$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{... | By the definition of [[Definition:Real Number|real number]]:
:$\forall \epsilon \in \R_{>0}: \exists t \in \N: \forall i > t: \size {x_i - x} < \epsilon$
{{explain|In order for this line to make sense, it needs the definition (or at least the relevant part) extracted and posted here, so the context of the convergence... | Inverse for Real Multiplication | https://proofwiki.org/wiki/Inverse_for_Real_Multiplication | https://proofwiki.org/wiki/Inverse_for_Real_Multiplication | [
"Real Multiplication",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Multiplication/Real Numbers"
] | [
"Definition:Real Number",
"Triangle Inequality/Real Numbers",
"Definition:Heaviside Step Function",
"Definition:Inverse (Abstract Algebra)/Inverse"
] |
proofwiki-3177 | Complex Multiplication Identity is One | Let $\C_{\ne 0}$ be the set of complex numbers without zero.
The identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$. | {{begin-eqn}}
{{eqn | l = \paren {x + i y} \paren {1 + 0 i}
| r = \paren {x \cdot 1 - y \cdot 0} + i \paren {x \cdot 0 + y \cdot 1}
| c =
}}
{{eqn | r = \paren {x + i y}
| c =
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn | l = \paren {1 + 0 i} \paren {x + i y}
| r = \paren {1 \cdot x - 0 ... | Let $\C_{\ne 0}$ be the set of [[Definition:Complex Number|complex numbers]] without [[Definition:Zero (Number)|zero]].
The [[Definition:Identity Element|identity element]] of $\struct {\C_{\ne 0}, \times}$ is the [[Definition:Complex Number|complex number]] $1 + 0 i$. | {{begin-eqn}}
{{eqn | l = \paren {x + i y} \paren {1 + 0 i}
| r = \paren {x \cdot 1 - y \cdot 0} + i \paren {x \cdot 0 + y \cdot 1}
| c =
}}
{{eqn | r = \paren {x + i y}
| c =
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn | l = \paren {1 + 0 i} \paren {x + i y}
| r = \paren {1 \cdot x -... | Complex Multiplication Identity is One | https://proofwiki.org/wiki/Complex_Multiplication_Identity_is_One | https://proofwiki.org/wiki/Complex_Multiplication_Identity_is_One | [
"Complex Multiplication",
"Examples of Identity Elements"
] | [
"Definition:Complex Number",
"Definition:Zero (Number)",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Complex Number"
] | [] |
proofwiki-3178 | Inverse for Complex Multiplication | Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $z^{-1}$ under the operation of complex multiplication:
:$\forall z \in \C_{\ne 0}: \exists z^{-1} \in \C_{\ne 0}: z \times z^{-1} = 1 + 0 i = z^{-1} \times z$
This inverse can be expressed as:
:$\dfrac 1 z := \dfrac {... | {{begin-eqn}}
{{eqn | l = \paren {x + i y} \frac {x - i y} {x^2 + y^2}
| r = \frac {\paren {x \cdot x - y \cdot \paren {-y} } + i \paren {x \cdot \paren {-y} + x \cdot y} } {x^2 + y^2}
| c =
}}
{{eqn | r = \frac {\paren {x^2 + y^2} + 0 i} {x^2 + y^2}
| c =
}}
{{eqn | r = 1 + 0 i
| c =
}}
{{en... | Each element $z = x + i y$ of the [[Definition:Set|set]] of [[Definition:Complex Number|non-zero complex numbers]] $\C_{\ne 0}$ has an [[Definition:Inverse Element|inverse element]] $z^{-1}$ under the operation of [[Definition:Complex Multiplication|complex multiplication]]:
:$\forall z \in \C_{\ne 0}: \exists z^{-1} \... | {{begin-eqn}}
{{eqn | l = \paren {x + i y} \frac {x - i y} {x^2 + y^2}
| r = \frac {\paren {x \cdot x - y \cdot \paren {-y} } + i \paren {x \cdot \paren {-y} + x \cdot y} } {x^2 + y^2}
| c =
}}
{{eqn | r = \frac {\paren {x^2 + y^2} + 0 i} {x^2 + y^2}
| c =
}}
{{eqn | r = 1 + 0 i
| c =
}}
{{en... | Inverse for Complex Multiplication | https://proofwiki.org/wiki/Inverse_for_Complex_Multiplication | https://proofwiki.org/wiki/Inverse_for_Complex_Multiplication | [
"Inverse for Complex Multiplication",
"Complex Multiplication",
"Examples of Inverse Elements"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Multiplication/Complex Numbers",
"Definition:Complex Conjugate"
] | [
"Definition:Complex Conjugate",
"Definition:Complex Modulus",
"Modulus in Terms of Conjugate"
] |
proofwiki-3179 | Additive Group of Integers is Normal Subgroup of Complex | Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\C, +}$ be the additive group of complex numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$. | From Additive Group of Integers is Normal Subgroup of Reals, $\struct {\Z, +} \lhd \struct {\R, +}$.
From Additive Group of Reals is Subgroup of Complex, $\struct {\R, +} \lhd \struct {\C, +}$.
Thus $\struct {\Z, +} \le \struct {\C, +}$.
From Complex Numbers under Addition form Infinite Abelian Group, $\struct {\C, +}$... | Let $\struct {\Z, +}$ be the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Then $\struct {\Z, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\C, +}$. | From [[Additive Group of Integers is Normal Subgroup of Reals]], $\struct {\Z, +} \lhd \struct {\R, +}$.
From [[Additive Group of Reals is Subgroup of Complex]], $\struct {\R, +} \lhd \struct {\C, +}$.
Thus $\struct {\Z, +} \le \struct {\C, +}$.
From [[Complex Numbers under Addition form Infinite Abelian Group]], $\... | Additive Group of Integers is Normal Subgroup of Complex | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Complex | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Complex | [
"Additive Group of Integers",
"Additive Group of Complex Numbers",
"Examples of Normal Subgroups"
] | [
"Definition:Additive Group of Integers",
"Definition:Additive Group of Complex Numbers",
"Definition:Normal Subgroup"
] | [
"Additive Group of Integers is Normal Subgroup of Reals",
"Additive Group of Reals is Subgroup of Complex",
"Complex Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Category:Additive Group of Integers",
"Category:Additive Group of ... |
proofwiki-3180 | Additive Group of Rationals is Normal Subgroup of Complex | Let $\struct {\Q, +}$ be the additive group of rational numbers.
Let $\struct {\C, +}$ be the additive group of complex numbers.
Then $\struct {\Q, +}$ is a normal subgroup of $\struct {\C, +}$. | From Additive Group of Rationals is Normal Subgroup of Reals, $\struct {\Q, +} \lhd \struct {\R, +}$.
From Additive Group of Reals is Normal Subgroup of Complex, $\struct {\R, +} \lhd \struct {\C, +}$.
Thus $\struct {\Q, +} \le \struct {\C, +}$.
From Complex Numbers under Addition form Infinite Abelian Group, $\struct ... | Let $\struct {\Q, +}$ be the [[Definition:Additive Group of Rational Numbers|additive group of rational numbers]].
Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Then $\struct {\Q, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\... | From [[Additive Group of Rationals is Normal Subgroup of Reals]], $\struct {\Q, +} \lhd \struct {\R, +}$.
From [[Additive Group of Reals is Normal Subgroup of Complex]], $\struct {\R, +} \lhd \struct {\C, +}$.
Thus $\struct {\Q, +} \le \struct {\C, +}$.
From [[Complex Numbers under Addition form Infinite Abelian Gro... | Additive Group of Rationals is Normal Subgroup of Complex | https://proofwiki.org/wiki/Additive_Group_of_Rationals_is_Normal_Subgroup_of_Complex | https://proofwiki.org/wiki/Additive_Group_of_Rationals_is_Normal_Subgroup_of_Complex | [
"Additive Group of Rational Numbers",
"Additive Group of Complex Numbers",
"Examples of Normal Subgroups"
] | [
"Definition:Additive Group of Rational Numbers",
"Definition:Additive Group of Complex Numbers",
"Definition:Normal Subgroup"
] | [
"Additive Group of Rationals is Normal Subgroup of Reals",
"Additive Group of Reals is Normal Subgroup of Complex",
"Complex Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Category:Additive Group of Rational Numbers",
"Category:Ad... |
proofwiki-3181 | Multiplicative Group of Rationals is Normal Subgroup of Complex | Let $\struct {\Q, \times}$ be the multiplicative group of rational numbers.
Let $\struct {\C, \times}$ be the multiplicative group of complex numbers.
Then $\struct {\Q, \times}$ is a normal subgroup of $\struct {\C, \times}$. | From Multiplicative Group of Rationals is Normal Subgroup of Reals, $\struct {\Q, \times} \lhd \struct {\R, \times}$.
From Multiplicative Group of Reals is Normal Subgroup of Complex, $\struct {\R, \times} \lhd \struct {\C, \times}$.
Thus $\struct {\Q, \times} \le \struct {\C, \times}$.
From Non-Zero Complex Numbers un... | Let $\struct {\Q, \times}$ be the [[Definition:Multiplicative Group of Rational Numbers|multiplicative group of rational numbers]].
Let $\struct {\C, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Then $\struct {\Q, \times}$ is a [[Definition:Normal Su... | From [[Multiplicative Group of Rationals is Normal Subgroup of Reals]], $\struct {\Q, \times} \lhd \struct {\R, \times}$.
From [[Multiplicative Group of Reals is Normal Subgroup of Complex]], $\struct {\R, \times} \lhd \struct {\C, \times}$.
Thus $\struct {\Q, \times} \le \struct {\C, \times}$.
From [[Non-Zero Compl... | Multiplicative Group of Rationals is Normal Subgroup of Complex | https://proofwiki.org/wiki/Multiplicative_Group_of_Rationals_is_Normal_Subgroup_of_Complex | https://proofwiki.org/wiki/Multiplicative_Group_of_Rationals_is_Normal_Subgroup_of_Complex | [
"Multiplicative Group of Rational Numbers",
"Multiplicative Group of Complex Numbers",
"Examples of Normal Subgroups"
] | [
"Definition:Multiplicative Group of Rational Numbers",
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Normal Subgroup"
] | [
"Multiplicative Group of Rationals is Normal Subgroup of Reals",
"Multiplicative Group of Reals is Normal Subgroup of Complex",
"Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Category:Multiplicative Group of ... |
proofwiki-3182 | Rational Numbers form Subfield of Complex Numbers | Let $\struct {\Q, +, \times}$ denote the field of rational numbers.
Let $\struct {\C, +, \times}$ denote the field of complex numbers.
$\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$. | From Rational Numbers form Subfield of Real Numbers, $\struct {\Q, +, \times}$ is a subfield of $\struct {\R, + \times}$.
From Real Numbers form Subfield of Complex Numbers, $\struct {\R, +, \times}$ is a subfield of $\struct {\C, + \times}$.
Thus from Subfield of Subfield is Subfield $\struct {\Q, +, \times}$ is a sub... | Let $\struct {\Q, +, \times}$ denote the [[Definition:Field of Rational Numbers|field of rational numbers]].
Let $\struct {\C, +, \times}$ denote the [[Definition:Field of Complex Numbers|field of complex numbers]].
$\struct {\Q, +, \times}$ is a [[Definition:Subfield|subfield]] of $\struct {\C, +, \times}$. | From [[Rational Numbers form Subfield of Real Numbers]], $\struct {\Q, +, \times}$ is a [[Definition:Subfield|subfield]] of $\struct {\R, + \times}$.
From [[Real Numbers form Subfield of Complex Numbers]], $\struct {\R, +, \times}$ is a [[Definition:Subfield|subfield]] of $\struct {\C, + \times}$.
Thus from [[Subfie... | Rational Numbers form Subfield of Complex Numbers | https://proofwiki.org/wiki/Rational_Numbers_form_Subfield_of_Complex_Numbers | https://proofwiki.org/wiki/Rational_Numbers_form_Subfield_of_Complex_Numbers | [
"Rational Numbers",
"Complex Numbers",
"Examples of Subfields"
] | [
"Definition:Field of Rational Numbers",
"Definition:Field of Complex Numbers",
"Definition:Subfield"
] | [
"Rational Numbers form Subfield of Real Numbers",
"Definition:Subfield",
"Real Numbers form Subfield of Complex Numbers",
"Definition:Subfield",
"Subfield of Subfield is Subfield",
"Definition:Subfield"
] |
proofwiki-3183 | Subfield of Subfield is Subfield | Let $R$ be a ring with unity.
Let $K_1, K_2$ be fields, such that:
: $K_1$ is a subfield of $R$
: $K_2$ is a subfield of $K_1$
Then $K_2$ is a subfield of $R$. | Let $K_1$ be a subfield of $R$ and $K_2$ be a subfield of $K_1$.
Then by definition:
:$K_1 \subseteq R$
:$K_2 \subseteq K_1$
From Subset Relation is Transitive it follows that $K_2 \subseteq R$
So by definition $K_2$ is a subfield of $R$.
{{qed}}
Category:Subfields
g8qj1gxkqsgvlury7jpgaahfpiooiz9 | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $K_1, K_2$ be [[Definition:Field (Abstract Algebra)|fields]], such that:
: $K_1$ is a [[Definition:Subfield|subfield]] of $R$
: $K_2$ is a [[Definition:Subfield|subfield]] of $K_1$
Then $K_2$ is a [[Definition:Subfield|subfield]] of $R$. | Let $K_1$ be a [[Definition:Subfield|subfield]] of $R$ and $K_2$ be a [[Definition:Subfield|subfield]] of $K_1$.
Then by definition:
:$K_1 \subseteq R$
:$K_2 \subseteq K_1$
From [[Subset Relation is Transitive]] it follows that $K_2 \subseteq R$
So by definition $K_2$ is a [[Definition:Subfield|subfield]] of $R$.
{{... | Subfield of Subfield is Subfield | https://proofwiki.org/wiki/Subfield_of_Subfield_is_Subfield | https://proofwiki.org/wiki/Subfield_of_Subfield_is_Subfield | [
"Subfields"
] | [
"Definition:Ring with Unity",
"Definition:Field (Abstract Algebra)",
"Definition:Subfield",
"Definition:Subfield",
"Definition:Subfield"
] | [
"Definition:Subfield",
"Definition:Subfield",
"Subset Relation is Transitive",
"Definition:Subfield",
"Category:Subfields"
] |
proofwiki-3184 | Nakayama's Lemma | Let $A$ be a commutative ring with unity.
Let $M$ be a finitely generated $A$-module.
Let $\map {\operatorname{Jac} } A$ be the Jacobson radical of $A$.
Let $\mathfrak a \subseteq \map {\operatorname{Jac} } A$ be an ideal of $A$.
Suppose $\mathfrak a M = M$.
Then:
:$M = 0$ | We induct on the number of generators of $M$.
;Basis for the Induction
Let $M$ have a single generator $m_1 \in M$.
Then $\mathfrak a m_1 = M$.
So:
:$m_1 \in \mathfrak a m_1$
That is:
:$m_1 = a m_1$
for some $a \in \mathfrak a$.
By Characterisation of Jacobson Radical, $1 - a$ is a unit in $A$.
So:
:$\paren {1 - a}^{-... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Finitely Generated Module|finitely generated $A$-module]].
Let $\map {\operatorname{Jac} } A$ be the [[Definition:Jacobson Radical|Jacobson radical]] of $A$.
Let $\mathfrak a \subseteq \map {\operatorname... | We [[Principle of Mathematical Induction|induct]] on the number of [[Definition:Generator of Module|generators]] of $M$.
;Basis for the Induction
Let $M$ have a single [[Definition:Generator of Module|generator]] $m_1 \in M$.
Then $\mathfrak a m_1 = M$.
So:
:$m_1 \in \mathfrak a m_1$
That is:
:$m_1 = a m_1$
for ... | Nakayama's Lemma/Proof 1 | https://proofwiki.org/wiki/Nakayama's_Lemma | https://proofwiki.org/wiki/Nakayama's_Lemma/Proof_1 | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Finitely Generated Module",
"Definition:Jacobson Radical",
"Definition:Ideal of Ring"
] | [
"Principle of Mathematical Induction",
"Definition:Generator of Module",
"Definition:Generator of Module",
"Characterisation of Jacobson Radical",
"Definition:Unit of Ring",
"Definition:Generator of Module",
"Characterisation of Jacobson Radical",
"Definition:Unit of Ring",
"Definition:Generator of ... |
proofwiki-3185 | Nakayama's Lemma | Let $A$ be a commutative ring with unity.
Let $M$ be a finitely generated $A$-module.
Let $\map {\operatorname{Jac} } A$ be the Jacobson radical of $A$.
Let $\mathfrak a \subseteq \map {\operatorname{Jac} } A$ be an ideal of $A$.
Suppose $\mathfrak a M = M$.
Then:
:$M = 0$ | Let $\phi : M \to M$ be the identity mapping on $M$, i.e.:
:$\forall x \in M : \map \phi x = x$
Since $\mathfrak a M = M$ {{hypothesis}}, $\phi$ is an endomorphism of $M$ such that:
:$\map \phi M \subseteq a M$
By Cayley-Hamilton Theorem, there exist $a_0, \ldots , a_{n-1} \in \mathfrak a$ such that:
:$(1):\quad \phi^n... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Finitely Generated Module|finitely generated $A$-module]].
Let $\map {\operatorname{Jac} } A$ be the [[Definition:Jacobson Radical|Jacobson radical]] of $A$.
Let $\mathfrak a \subseteq \map {\operatorname... | Let $\phi : M \to M$ be the [[Definition:Identity Mapping|identity mapping]] on $M$, i.e.:
:$\forall x \in M : \map \phi x = x$
Since $\mathfrak a M = M$ {{hypothesis}}, $\phi$ is an [[Definition:Endomorphism|endomorphism]] of $M$ such that:
:$\map \phi M \subseteq a M$
By [[Cayley-Hamilton Theorem for Finitely Gener... | Nakayama's Lemma/Proof 2 | https://proofwiki.org/wiki/Nakayama's_Lemma | https://proofwiki.org/wiki/Nakayama's_Lemma/Proof_2 | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Finitely Generated Module",
"Definition:Jacobson Radical",
"Definition:Ideal of Ring"
] | [
"Definition:Identity Mapping",
"Definition:Endomorphism",
"Cayley-Hamilton Theorem/Finitely Generated Module",
"Characterisation of Jacobson Radical",
"Definition:Unit of Ring"
] |
proofwiki-3186 | Characterisation of Jacobson Radical | Let $A$ be a commutative ring with unity.
Let $A^\times$ be the group of units of $A$.
Let $\map {\operatorname {Jac} } A$ be the Jacobson radical of $A$.
Then:
:$\map {\operatorname {Jac} } A = \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$
where $1_A$ is the unity of $A$. | Recall the definition of Jacobson radical of $A$:
:$\map {\operatorname {Jac} } A$ is the intersection of all maximal ideals of $R$. | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $A^\times$ be the [[Definition:Group of Units of Ring|group of units]] of $A$.
Let $\map {\operatorname {Jac} } A$ be the [[Definition:Jacobson Radical|Jacobson radical]] of $A$.
Then:
:$\map {\operatorname {Jac} } A = \set {a... | Recall the definition of [[Definition:Jacobson Radical|Jacobson radical]] of $A$:
:$\map {\operatorname {Jac} } A$ is the [[Definition:Set Intersection|intersection]] of all [[Definition:Maximal Ideal of Ring|maximal ideals]] of $R$. | Characterisation of Jacobson Radical | https://proofwiki.org/wiki/Characterisation_of_Jacobson_Radical | https://proofwiki.org/wiki/Characterisation_of_Jacobson_Radical | [
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Group of Units/Ring",
"Definition:Jacobson Radical",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Definition:Jacobson Radical",
"Definition:Set Intersection",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring"
] |
proofwiki-3187 | Prime Ideal iff Quotient Ring is Integral Domain | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $J$ be an ideal of $R$.
Then $J$ is a prime ideal of $R$ {{iff}} the quotient ring $R / J$ is an integral domain. | Since $J \subset R$, it follows from:
:Quotient Ring of Commutative Ring is Commutative
and:
:Quotient Ring of Ring with Unity is Ring with Unity
that $R / J$ is a commutative ring with unity.
Let $0_{R / J}$ be the zero of $R / J$. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Then $J$ is a [[Definition:Prime Ideal of Commutative and Unitary Ring|prime ideal]] of $R$ {{iff}} the [[Definition:Quotient Ring|quotient ring]] $R / J$... | Since $J \subset R$, it follows from:
:[[Quotient Ring of Commutative Ring is Commutative]]
and:
:[[Quotient Ring of Ring with Unity is Ring with Unity]]
that $R / J$ is a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $0_{R / J}$ be the [[Definition:Ring Zero|zero]] of $R / J$. | Prime Ideal iff Quotient Ring is Integral Domain | https://proofwiki.org/wiki/Prime_Ideal_iff_Quotient_Ring_is_Integral_Domain | https://proofwiki.org/wiki/Prime_Ideal_iff_Quotient_Ring_is_Integral_Domain | [
"Quotient Rings",
"Prime Ideals of Rings",
"Integral Domains"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring",
"Definition:Quotient Ring",
"Definition:Integral Domain"
] | [
"Quotient Ring of Commutative Ring is Commutative",
"Quotient Ring of Ring with Unity is Ring with Unity",
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero",
"Definition:Ring Zero"
] |
proofwiki-3188 | Maximal Ideal of Commutative and Unitary Ring is Prime Ideal | Let $R$ be a commutative ring with unity.
Let $M$ be a maximal ideal of $R$.
Then $M$ is a prime ideal of $R$. | From Maximal Ideal iff Quotient Ring is Field:
:the quotient ring $R / M$ is a field.
It follows from Field is Integral Domain that $R / M$ is an integral domain.
Finally it follows from Prime Ideal iff Quotient Ring is Integral Domain that $M$ is a prime ideal.
{{qed}} | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $M$ be a [[Definition:Maximal Ideal of Ring|maximal ideal]] of $R$.
Then $M$ is a [[Definition:Prime Ideal of Commutative and Unitary Ring|prime ideal]] of $R$. | From [[Maximal Ideal iff Quotient Ring is Field]]:
:the [[Definition:Quotient Ring|quotient ring]] $R / M$ is a [[Definition:Field (Abstract Algebra)|field]].
It follows from [[Field is Integral Domain]] that $R / M$ is an [[Definition:Integral Domain|integral domain]].
Finally it follows from [[Prime Ideal iff Quoti... | Maximal Ideal of Commutative and Unitary Ring is Prime Ideal | https://proofwiki.org/wiki/Maximal_Ideal_of_Commutative_and_Unitary_Ring_is_Prime_Ideal | https://proofwiki.org/wiki/Maximal_Ideal_of_Commutative_and_Unitary_Ring_is_Prime_Ideal | [
"Maximal Ideals of Rings",
"Prime Ideals of Commutative and Unitary Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring"
] | [
"Maximal Ideal iff Quotient Ring is Field",
"Definition:Quotient Ring",
"Definition:Field (Abstract Algebra)",
"Field is Integral Domain",
"Definition:Integral Domain",
"Prime Ideal iff Quotient Ring is Integral Domain",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring"
] |
proofwiki-3189 | Radical Ideal iff Quotient Ring is Reduced | Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.
Let $J$ be an ideal of $R$.
Then $J$ is a radical ideal {{iff}} the quotient ring $R / J$ is a reduced ring. | Since $J \subset R$, it follows from:
:Quotient Ring of Commutative Ring is Commutative
and:
: Quotient Ring of Ring with Unity is Ring with Unity
that $R / J$ is a commutative ring with unity.
Let $0_{R / J}$ be the zero of $R / J$. | Let $\left({R, +, \circ}\right)$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Then $J$ is a [[Definition:Radical Ideal of Ring|radical ideal]] {{iff}} the [[Definition:Quotient Ring|quotient ring]] $R / J$ is a [[Definition:Red... | Since $J \subset R$, it follows from:
:[[Quotient Ring of Commutative Ring is Commutative]]
and:
: [[Quotient Ring of Ring with Unity is Ring with Unity]]
that $R / J$ is a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $0_{R / J}$ be the [[Definition:Ring Zero|zero]] of $R / J$. | Radical Ideal iff Quotient Ring is Reduced | https://proofwiki.org/wiki/Radical_Ideal_iff_Quotient_Ring_is_Reduced | https://proofwiki.org/wiki/Radical_Ideal_iff_Quotient_Ring_is_Reduced | [
"Quotient Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Radical Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Reduced Ring"
] | [
"Quotient Ring of Commutative Ring is Commutative",
"Quotient Ring of Ring with Unity is Ring with Unity",
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero"
] |
proofwiki-3190 | Universal Property of Polynomial Ring/Free Monoid on Set | Let $R, S$ be commutative and unitary rings.
Let $\family {s_j}_{j \mathop \in J}$ be an indexed family of elements of $S$.
Let $\psi: R \to S$ be a ring homomorphism.
Let $R \sqbrk {\set {X_j: j \in J} }$ be a polynomial ring.
Then there exists a unique evaluation homomorphism $\phi: R \sqbrk {\set {X_j: j \in J} } \t... | Let $Z$ be the set of all multiindices indexed by $J$.
Let $k_j$ be the $j$th component of a multiindex $k$.
Let $\ds f = \sum_{k \mathop \in Z} a_k \prod_{j \mathop \in J} X_j^{k_j}$ be a polynomial over $R$.
Define:
:$\ds \map \phi f = \sum_{k \mathop \in Z} \map \psi {a_k} \prod_{j \mathop \in J} s_j^{k_j}$
It is cl... | Let $R, S$ be [[Definition:Commutative and Unitary Ring|commutative and unitary rings]].
Let $\family {s_j}_{j \mathop \in J}$ be an [[Definition:Indexed Family|indexed family]] of elements of $S$.
Let $\psi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $R \sqbrk {\set {X_j: j \in J} }$ be a... | Let $Z$ be the set of all [[Definition:Multiindex|multiindices]] indexed by $J$.
Let $k_j$ be the $j$th component of a [[Definition:Multiindex|multiindex]] $k$.
Let $\ds f = \sum_{k \mathop \in Z} a_k \prod_{j \mathop \in J} X_j^{k_j}$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] over $R$.
Define:
... | Universal Property of Polynomial Ring/Free Monoid on Set | https://proofwiki.org/wiki/Universal_Property_of_Polynomial_Ring/Free_Monoid_on_Set | https://proofwiki.org/wiki/Universal_Property_of_Polynomial_Ring/Free_Monoid_on_Set | [
"Polynomial Theory",
"Universal Properties"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Indexing Set/Family",
"Definition:Ring Homomorphism",
"Definition:Polynomial Ring",
"Definition:Polynomial Evaluation Homomorphism",
"Definition:Extension of Mapping"
] | [
"Definition:Multiindex",
"Definition:Multiindex",
"Definition:Polynomial over Ring",
"Axiom:Ring Axioms",
"Axiom:Ring Axioms",
"Definition:Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract A... |
proofwiki-3191 | Set of Monomials is Closed Under Multiplication | Let $M$ be the set of all monomials on the set $\set {X_j: j \in J}$, with multiplication $\circ$ defined by:
:$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$
Then $M$ is closed under $\circ$. | Let $\ds m_1 = \prod_{j \mathop \in J} X_j^{k_j}, m_2 = \prod_{j \mathop \in J} X_j^{k_j'}$ be two monomials.
Their product is:
:$\ds m_1 \circ m_2 = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$
If $k_j + k_j' \ne 0$ then either $k_j \ne 0$ or $k_j' \ne 0$ (or both are nonzero).
Therefore if $k_j + k_j' \ne 0$ f... | Let $M$ be the set of all [[Definition:Monomial|monomials]] on the set $\set {X_j: j \in J}$, with multiplication $\circ$ defined by:
:$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$
Then $M$ is closed under $\c... | Let $\ds m_1 = \prod_{j \mathop \in J} X_j^{k_j}, m_2 = \prod_{j \mathop \in J} X_j^{k_j'}$ be two [[Definition:Monomial|monomials]].
Their product is:
:$\ds m_1 \circ m_2 = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$
If $k_j + k_j' \ne 0$ then either $k_j \ne 0$ or $k_j' \ne 0$ (or both are nonzero).
There... | Set of Monomials is Closed Under Multiplication | https://proofwiki.org/wiki/Set_of_Monomials_is_Closed_Under_Multiplication | https://proofwiki.org/wiki/Set_of_Monomials_is_Closed_Under_Multiplication | [
"Monomials"
] | [
"Definition:Monomial"
] | [
"Definition:Monomial",
"Definition:Monomial",
"Category:Monomials"
] |
proofwiki-3192 | Free Commutative Monoid is Commutative Monoid | The free commutative monoid on a set $\family {X_j: j \in J}$ is a commutative monoid. | Let $M$ be the set of all monomials on the indexed set $\family {X_j: j \in J}$.
We are required to show that the following properties hold:
{{begin-axiom}}
{{axiom | q = \forall m_1, m_2 \in M
| m = m_1 \circ m_2 \in M
| t = Closure
}}
{{axiom | q = \forall m_1, m_2, m)3 \in M
| m = \paren {m_1... | The [[Definition:Free Commutative Monoid|free commutative monoid]] on a set $\family {X_j: j \in J}$ is a [[Definition:Commutative Monoid|commutative monoid]]. | Let $M$ be the set of all [[Definition:Monomial|monomials]] on the indexed set $\family {X_j: j \in J}$.
We are required to show that the following properties hold:
{{begin-axiom}}
{{axiom | q = \forall m_1, m_2 \in M
| m = m_1 \circ m_2 \in M
| t = [[Definition:Closed Algebraic Structure|Closure]]
}}... | Free Commutative Monoid is Commutative Monoid | https://proofwiki.org/wiki/Free_Commutative_Monoid_is_Commutative_Monoid | https://proofwiki.org/wiki/Free_Commutative_Monoid_is_Commutative_Monoid | [
"Examples of Commutative Monoids",
"Polynomial Theory"
] | [
"Definition:Free Commutative Monoid",
"Definition:Commutative Monoid"
] | [
"Definition:Monomial",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation",
"Definition:Commutative/Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiindex",
"Definition:Monomial",
"Definition:Multiindex",
"Definitio... |
proofwiki-3193 | Polynomial Addition is Associative | Addition of polynomials is an associative operation. | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
:$\ds h = \sum_{k \matho... | [[Definition:Addition of Polynomial Forms|Addition of polynomials]] is an [[Definition:Associative Operation|associative]] operation. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\set {X_j: j \in J}$ be a [[Definition:Set|set]] of [[Definition:Indeterminate (Polynomial Theory)|indeterminates]].
Let $Z$ be the [[Definition:Set|set]] of all [[Definition:Multiindex|multiindices]] indexe... | Polynomial Addition is Associative | https://proofwiki.org/wiki/Polynomial_Addition_is_Associative | https://proofwiki.org/wiki/Polynomial_Addition_is_Associative | [
"Polynomial Theory"
] | [
"Definition:Polynomial Addition/Polynomial Forms",
"Definition:Associative Operation"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Set",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Set",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Associative Operation",
"Category:Polynomial Theory"
] |
proofwiki-3194 | Null Polynomial is Additive Identity | The set of polynomial forms has an additive identity.
{{explain|Context}} | Let $\struct {R, +, \circ}$ be a commutative ring with unity with zero $0_R$.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
be an arbitrary polynomial form in the indeterminates $\s... | The [[Definition:Set|set]] of [[Definition:Polynomial Form|polynomial forms]] has an [[Definition:Identity Element|additive identity]].
{{explain|Context}} | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] with [[Definition:Ring Zero|zero]] $0_R$.
Let $\set {X_j: j \in J}$ be a [[Definition:Set|set]] of [[Definition:Indeterminate (Polynomial Theory)|indeterminates]].
Let $Z$ be the [[Definition:Set|set]] of all [[... | Null Polynomial is Additive Identity | https://proofwiki.org/wiki/Null_Polynomial_is_Additive_Identity | https://proofwiki.org/wiki/Null_Polynomial_is_Additive_Identity | [
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Polynomial over Ring as Function on Free Monoid on Set",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero",
"Definition:Set",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Set",
"Definition:Multiindex",
"Definition:Polynomial over Ring as Function on Free Monoid on Set",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Nul... |
proofwiki-3195 | Polynomial Addition is Commutative | Addition of polynomials is commutative. | Let $\struct {R, +, \circ}$ be a commutative ring with unity with zero $0_R$.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
be arbitra... | [[Definition:Addition of Polynomial Forms|Addition of polynomials]] is [[Definition:Commutative Operation|commutative]]. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] with [[Definition:Ring Zero|zero]] $0_R$.
Let $\set {X_j: j \in J}$ be a set of [[Definition:Indeterminate (Polynomial Theory)|indeterminates]].
Let $Z$ be the set of all [[Definition:Multiindex|multiindices]] ind... | Polynomial Addition is Commutative | https://proofwiki.org/wiki/Polynomial_Addition_is_Commutative | https://proofwiki.org/wiki/Polynomial_Addition_is_Commutative | [
"Polynomial Theory"
] | [
"Definition:Polynomial Addition/Polynomial Forms",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Commutative/Operation",
"Definition:Polynomial",
"Definition:Polynomial Addition/Po... |
proofwiki-3196 | Multiplication of Polynomials is Associative | Multiplication of polynomials is associative. | Let $\struct {R, +, \circ}$ be a commutative ring with unity with zero $0_R$.
To improve readability of the expressions used, we will write the ring product $\circ$ in multiplicative notation.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
... | [[Definition:Multiplication of Polynomial Forms|Multiplication of polynomials]] is [[Definition:Associative Operation|associative]]. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] with [[Definition:Ring Zero|zero]] $0_R$.
To improve readability of the expressions used, we will write the [[Definition:Ring Product|ring product]] $\circ$ in [[Definition:Multiplicative Notation|multiplicative no... | Multiplication of Polynomials is Associative | https://proofwiki.org/wiki/Multiplication_of_Polynomials_is_Associative | https://proofwiki.org/wiki/Multiplication_of_Polynomials_is_Associative | [
"Polynomial Theory"
] | [
"Definition:Multiplication of Polynomials/Polynomial Forms",
"Definition:Associative Operation"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring Zero",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Multiplicative Notation",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Polynomial... |
proofwiki-3197 | Polynomials Contain Multiplicative Identity | The set of polynomials has a multiplicative identity. | Let $\struct {R, +, \circ}$ be a commutative ring with unity with multiplicative identity $1_R$ and additive identity $0_R$.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\left\{{X_j: j \in J}\right\}$.
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
be... | The [[Definition:Set|set]] of [[Definition:Polynomial|polynomials]] has a [[Definition:Multiplicative Identity|multiplicative identity]]. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] with [[Definition:Multiplicative Identity|multiplicative identity]] $1_R$ and [[Definition:Additive Identity|additive identity]] $0_R$.
Let $\set {X_j: j \in J}$ be a [[Definition:Set|set]] of [[Definition:Indeterm... | Polynomials Contain Multiplicative Identity | https://proofwiki.org/wiki/Polynomials_Contain_Multiplicative_Identity | https://proofwiki.org/wiki/Polynomials_Contain_Multiplicative_Identity | [
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Polynomial",
"Definition:Multiplicative Identity"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Multiplicative Identity",
"Definition:Field Zero",
"Definition:Set",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Set",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multip... |
proofwiki-3198 | Multiplication of Polynomials is Commutative | Multiplication of polynomials is commutative. | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
be arbitrary polynomials i... | [[Definition:Polynomial Multiplication|Multiplication of polynomials]] is [[Definition:Commutative Operation|commutative]]. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\set {X_j: j \in J}$ be a set of [[Definition:Indeterminate (Polynomial Theory)|indeterminates]].
Let $Z$ be the set of all [[Definition:Multiindex|multiindices]] indexed by $\set {X_j: j \in J}$.
Let:
:$\... | Multiplication of Polynomials is Commutative | https://proofwiki.org/wiki/Multiplication_of_Polynomials_is_Commutative | https://proofwiki.org/wiki/Multiplication_of_Polynomials_is_Commutative | [
"Polynomial Theory"
] | [
"Definition:Multiplication of Polynomials",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multiindex",
"Definition:Commutative/Operation",
"Definition:Bound Variable",
"Definition:Polynomial",
"D... |
proofwiki-3199 | Multiplication of Polynomials Distributes over Addition | Multiplication of polynomials is left- and right- distributive over addition. | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $\set {X_j: j \in J}$ be a set of indeterminates.
Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.
Let
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
:$\ds h = \sum_{k \mathop... | Multiplication of [[Definition:Polynomial|polynomials]] is [[Definition:Left Distributive Operation|left]]- and [[Definition:Right Distributive Operation|right]]- [[Definition:Distributive Operation|distributive]] over addition. | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\set {X_j: j \in J}$ be a [[Definition:Set|set]] of [[Definition:Indeterminate (Polynomial Theory)|indeterminates]].
Let $Z$ be the set of all [[Definition:Multiindex|multiindices]] indexed by $\set {X_j: j ... | Multiplication of Polynomials Distributes over Addition | https://proofwiki.org/wiki/Multiplication_of_Polynomials_Distributes_over_Addition | https://proofwiki.org/wiki/Multiplication_of_Polynomials_Distributes_over_Addition | [
"Polynomial Theory",
"Examples of Distributive Operations"
] | [
"Definition:Polynomial",
"Definition:Distributive Operation/Left",
"Definition:Distributive Operation/Right",
"Definition:Distributive Operation"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Set",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Multiindex",
"Definition:Polynomial",
"Definition:Polynomial Ring/Indeterminate",
"Multiplication of Polynomials is Commutative",
"Definition:Distributive Operation/Left",
"Category:P... |
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