Theorems

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

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Existence and Uniqueness of Iterates of a Binary Operation
lemmalem:iterated-binary-operation-2026aAlgebra Set Theory
Let $X$ be a set and let $\ast$ be a binary operation on $X$ , that is, a map $\ast:X\times X\to X$ , whose value at $(x,y)$ is written $x\ast y$ . Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, let $n\in\mathbb{N}$ , let $[n]$ be the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T19:46:01.594654Z
The Complex Coordinate Space is a Complex Vector Space
lemmalem:cn-vector-space-2026aAlgebra Linear Algebra
Let $n$ be a natural number and let $\mathbb{C}^{n}$ be the complex coordinate space, with the componentwise operations of that definition. Then $\mathbb{C}^{n}$ with these operations is a complex vector space. Its zero vector is the $n$ -tuple $0_{\mathbb{C}^{n}}=(0,\dots,0)$ al…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:39:31.004970Z
The Complex Coordinate Space
definitiondef:complex-coordinate-space-cn-2026aAlgebra Linear Algebra
Let $n$ be a natural number and let $\mathbb{C}$ be the field of complex numbers. The complex coordinate space $\mathbb{C}^{n}$ is the set of all ordered $n$ -tuples $u=(u_{1},\dots,u_{n})$ of complex numbers, together with the operations defined componentwise by…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:39:27.733548Z
Properties of Finite Sums
lemmalem:finite-sum-properties-2026bAnalysis Algebra
Let $K$ be a field. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, ordered by the relation $\le$ of that definition, let $n\in\mathbb{N}$ , and let $[n]$ be the initial segment determined by $n$ , that is, the set of natural numbers $k$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:39:23.512020Z
Finite Sum Notation in a Field
definitiondef:finite-sum-field-2026bAlgebra
Let $K$ be a field, let $n$ be a natural number with successor map $S$ as in that definition, let $[n]$ be the initial segment determined by $n$ , and let $a:[n]\to K$ be a map, whose value at $k$ is written $a_{k}$ . Let $\sigma:[n]\to K$ be the map given by…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:39:16.510063Z
Linear Map
definitiondef:linear-map-2026aAlgebra Linear Algebra
Let $K$ be a field and let $V$ and $W$ be vector spaces over $K$ . A map $T:V\to W$ is linear if 1. $T(u+v)=T(u)+T(v)$ for all $u,v\in V$ ; 2. $T(\lambda v)=\lambda\,T(v)$ for all $\lambda\in K$ and $v\in V$ , where on the left-hand sides the operations are those of $V$ and on th…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:18:27.986235Z
Linear Subspace
definitiondef:linear-subspace-2026aAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ , and let $0_{V}$ be its zero vector. A subset $W$ of $V$ is a linear subspace of $V$ if 1. $0_{V}\in W$ ; 2. $u+v\in W$ for all $u,v\in W$ ; 3. $\lambda v\in W$ for all $\lambda\in K$ and $v\in W$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:18:25.698773Z
Elementary Identities in a Vector Space
lemmalem:vector-space-basic-identities-2026aAlgebra Linear Algebra
Let $K$ be a field and let $V$ be a vector space over $K$ , with the conditions 1-8 of that definition. Then the following hold. 1. (Uniqueness of the zero vector) There is exactly one element $0_{V}\in V$ with $v+0_{V}=v$ for every $v\in V$ . 2. (Uniqueness of additive inverses)…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T04:18:22.238849Z
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 $r$ …
+1 / -0flags 0verified 0no proof
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…
+1 / -0flags 0verified 0no proof
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…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:13:52.771063Z
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…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T03:13:15.027656Z
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…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:20:13.114162Z
A Group Homomorphism is Injective Exactly When its Kernel is Trivial
theoremthm:trivial-kernel-injective-2026aAlgebra
Let $(G,\ast_G)$ and $(H,\ast_H)$ be groups with identity elements $e_G$ and $e_H$ as in Uniqueness of the Identity Element and of Inverses in a Group, and let $\varphi:G\to H$ be a group homomorphism. Then $\varphi$ is injective if and only if its kernel satisfies…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:18:22.382673Z

Showing 41-60 of 76

Back Next

Existence and Uniqueness of Iterates of a Binary Operation

The Complex Coordinate Space is a Complex Vector Space

The Complex Coordinate Space

Properties of Finite Sums

Finite Sum Notation in a Field

Linear Map

Linear Subspace

Elementary Identities in a Vector Space

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

The Complex Numbers

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

A Group Homomorphism is Injective Exactly When its Kernel is Trivial