Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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Vector Space over a Field
definitiondef:vector-space-2026aAlgebra Linear Algebra
Let $K$ be a field, with additive identity $0$ and multiplicative identity $1$ . A vector space over $K$ is a set $V$ together with two operations: an addition, assigning to each pair $u,v$ of elements of $V$ an element $u+v$ of $V$ , and a scalar multiplication, assigning to each…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:18:18.874730Z
Canonical Form and Arithmetic of Complex Numbers
lemmalem:complex-canonical-form-2026aAnalysis Algebra
Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$ , and let $\mathbb{R}$ be the set of real numbers. Then the following hold. 1. (Identities and inverses of real numbers) The additive identity of $\mathbb{C}$ is the real number $0$ and the multiplicative id…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:53:48.202094Z
Existence and Uniqueness of the Complex Numbers
theoremthm:complex-numbers-existence-uniqueness-2026aAnalysis Algebra
Let $\mathbb{R}$ be the set of real numbers, with its addition, its multiplication and its order. Call a pair $(K,j)$ , consisting of a field $K$ and an element $j\in K$ , a complex pair if the following three conditions hold, where $+$ and $\cdot$ denote the addition and multipli…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:53:44.626610Z
Properties of Complex Conjugation and Modulus
lemmalem:complex-conjugate-modulus-properties-2026aAnalysis Algebra
Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$ , let $\mathbb{R}$ be the set of real numbers, and let $z,w\in\mathbb{C}$ . Sums, products and inverses are those of the field $\mathbb{C}$ , and $z-w$ abbreviates $z+(-w)$ ; for a real number $x$ , $2x$ abbrevia…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:14:39.810914Z
Modulus of a Complex Number
definitiondef:complex-modulus-2026aAnalysis Algebra
Let $\mathbb{C}$ be the field of complex numbers, let $z\in\mathbb{C}$ , and let $a=\operatorname{Re}z$ and $b=\operatorname{Im}z$ be its real and imaginary parts. The modulus of $z$ , written $|z|$ , is the unique real number $r$ with $0\le r$ and $r^{2}=a^{2}+b^{2}$ ; such an $r$ …
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Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:14:17.344611Z
Complex Conjugate
definitiondef:complex-conjugate-2026aAnalysis Algebra
Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$ , let $z\in\mathbb{C}$ , and let $\operatorname{Re}z$ and $\operatorname{Im}z$ be its real and imaginary parts. The complex conjugate of $z$ is the complex number…
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Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:14:14.024218Z
Real and Imaginary Parts of a Complex Number
definitiondef:complex-real-imaginary-part-2026aAnalysis Algebra
Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$ , and let $z\in\mathbb{C}$ . By claim 3 of Canonical Form and Arithmetic of Complex Numbers there is exactly one pair of real numbers $a,b$ with $z=a+bi$ . The real part of $z$ is the real number…
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Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:13:52.771063Z
Existence and Uniqueness of the Square Root of a Sum of Two Squares
lemmalem:sum-two-squares-square-root-2026aAnalysis
Let $a$ and $b$ be real numbers. Then $0\le a^{2}+b^{2}$ , and there is exactly one real number $r$ with $0\le r$ and $r^{2}=a^{2}+b^{2}$ .
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:13:50.505316Z
The Complex Numbers
definitiondef:complex-numbers-2026aAnalysis Algebra
Let $\mathbb{R}$ be the set of real numbers, with its addition and multiplication. The complex numbers are a field $\mathbb{C}$ , whose addition and multiplication are written $+$ and $\cdot$ (the product $z\cdot w$ being also written $zw$ ), together with a distinguished element…
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Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:13:15.027656Z
Divisibility of Natural Numbers
definitiondef:divides-natural-numbers-2026aNumber Theory Set Theory
Let $a,b\in\mathbb{N}$ , where $\mathbb{N}$ is the set of natural numbers with multiplication $\cdot$ as in that definition. We say that $a$ divides $b$ , written $a\mid b,$ if there exists $c\in\mathbb{N}$ with $b=a\cdot c$ .
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:59:06.135531Z
Finite Set
definitiondef:finite-set-2026aCombinatorics Set Theory
A set $X$ is called finite if $X=\emptyset$ or if $X$ has $n$ elements for some natural number $n$ .
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:49:22.295547Z
Properties of the Order on the Natural Numbers
lemmalem:order-natural-numbers-2026aNumber Theory Set Theory
Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ and successor map $S$ as in that definition, and let $<$ and $\le$ be the order relations of that definition. Then the following hold for all $j,k,m,n,p,t,y\in\mathbb{N}$ . 1. $m\le m$ ; if $m<n$ then $m\le n$ ; if…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:46:48.485008Z
Order on the Natural Numbers
definitiondef:order-natural-numbers-2026aNumber Theory Set Theory
Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ as in that definition, and let $m,n\in\mathbb{N}$ . We write $m<n$ if there exists $k\in\mathbb{N}$ with $n=m+k$ , and we write $m\le n$ if $m<n$ or $m=n$ . We also write $n>m$ for $m<n$ , and $n\ge m$ for…
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:46:35.547173Z
Arithmetic of Addition on the Natural Numbers
lemmalem:natural-number-addition-2026aNumber Theory Set Theory
Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ and successor map $S$ as in that definition. Then the following hold for all $a,b,c\in\mathbb{N}$ . 1. $a+1=S(a)$ and $1+a=S(a)$ . 2. $S(a)+b=S(a+b)$ . 3. (Associativity) $(a+b)+c=a+(b+c)$ . 4. (Commutativity)…
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:45:15.307174Z
Principle of Induction for the Natural Numbers
axiomaxiom:induction-natural-numbers-2026aLogic Set Theory
Let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ as in that definition. We take as an axiom the following principle of induction. If $A\subseteq\mathbb{N}$ satisfies 1. $1\in A$ , and 2. $S(n)\in A$ for every $n\in A$ , then $A=\mathbb{N}$ .
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:45:04.207379Z
Injectivity, Composition, and Restriction of Bijections
lemmalem:bijection-basic-2026aSet Theory
Let $X$ , $Y$ , and $Z$ be sets. 1. Every bijection $u:X\to Y$ is injective; that is, $u(x)=u(x')$ implies $x=x'$ for all $x,x'\in X$ . 2. If $u:X\to Y$ and $v:Y\to Z$ are bijections, then the map $w:X\to Z$ defined by $w(x)=v(u(x))$ is a bijection. 3. If $u:X\to Y$ is a bijection…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:32:42.811324Z
Lagrange's Theorem
theoremthm:lagrange-2026aAlgebra
Let $(G,\ast)$ be a group whose underlying set is finite, and let $H$ be a subgroup of $G$ . Then the following hold. 1. The sets $H$ and $G/H$ , where $G/H$ is the set of left cosets of $H$ in $G$ , are finite and nonempty, and $|G|=|H|\,[G:H],$ where $|G|$ is the order of $G$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:21:25.429311Z
Left Cosets Partition a Group and All Have the Same Cardinality
theoremthm:coset-partition-2026aAlgebra
Let $(G,\ast)$ be a group, written multiplicatively, with identity element $e_G$ and inverses $a^{-1}$ as in Uniqueness of the Identity Element and of Inverses in a Group, and let $H$ be a subgroup of $G$ . For $a\in G$ let $aH$ be the left coset determined by $a$ . Then for all…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:20:23.509507Z
Left Coset, Order of a Group, and Index of a Subgroup
definitiondef:left-coset-order-index-2026aAlgebra
Let $(G,\ast)$ be a group, written multiplicatively as $ab=a\ast b$ , and let $H$ be a subgroup of $G$ . For $a\in G$ , the left coset of $H$ determined by $a$ is the subset of $G$ given by $aH=\{ah:\ h\in H\}.$ The collection of all left cosets of $H$ in $G$ is denoted…
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:20:13.114162Z
Basic Properties of Finite Sets
lemmalem:finite-set-basic-2026aCombinatorics Set Theory
Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ , let $\le$ be the order on $\mathbb{N}$ , let $[n]$ denote the initial segment determined by $n$ , and let the notions number of elements $|X|$ and finite be as in those definitions. Then the following hold. 1.…
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:19:26.884800Z

Showing 1041-1060 of 1431

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Vector Space over a Field

Canonical Form and Arithmetic of Complex Numbers

Existence and Uniqueness of the Complex Numbers

Properties of Complex Conjugation and Modulus

Modulus of a Complex Number

Complex Conjugate

Real and Imaginary Parts of a Complex Number

Existence and Uniqueness of the Square Root of a Sum of Two Squares

The Complex Numbers

Divisibility of Natural Numbers

Finite Set

Properties of the Order on the Natural Numbers

Order on the Natural Numbers

Arithmetic of Addition on the Natural Numbers

Principle of Induction for the Natural Numbers

Injectivity, Composition, and Restriction of Bijections

Lagrange's Theorem

Left Cosets Partition a Group and All Have the Same Cardinality

Left Coset, Order of a Group, and Index of a Subgroup

Basic Properties of Finite Sets