Addition is commutative

Depends on

• Addition of natural numbers
• Left identity for addition
• Left successor law for addition
• The principle of mathematical induction

Used by

• If V = ⨁_i<n Uᵢ with every Uᵢ finite-dimensional, then V is finite-dimensional and dim_F V = ∑_i<n dim_F Uᵢ; in particular dim_F(U ⊕ W) = dim_F U + dim_F W Corollary
• A commutative monoid in which cancellation holds need not be a group: (ℕ, +) Counterexample
• ⌈ m/n ⌉ for naturals m and n ≥ 1: the least q ∈ ℕ with m ≤ n q Definition
• Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ Definition
• (ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1} Example
• For every n ∈ ℕ there are n consecutive composite integers: with N := ∏_j<n(j+2), each of N+2, …, N+n+1 is composite Example
• nℤ is a subgroup of (ℤ, +) for every n ∈ ℤ, and every subgroup of (ℤ, +) has this form Example
• The subsets of ℕ containing a tail form the Fréchet filter, and it is proper and not an ultrafilter Example
• A finite sum is unchanged by a permutation of its index range: ∑_k<n a_π(k) = ∑_k<n aₖ for every bijection π : n → n Lemma
• Convergence depends only on the tail Lemma
• Discreteness: σ(n) is the immediate successor Lemma
• Distributivity and the successor law for multiplication Lemma
• Exponent laws in a group: gᵐ⁺ⁿ = gᵐgⁿ and (gᵐ)ⁿ = gᵐⁿ for all m, n ∈ ℤ, and (gh)ⁿ = gⁿhⁿ **when g and h commute** Lemma
• Finite sums re-indexed along an injection, with a zero term deleted, and concatenated; and the closure properties of linear independence: an independent list is injective and never 0_V, its sublists are independent, a list is independent exactly when it is injective with linearly independent image, and every subset of a linearly independent set is linearly independent Lemma
• In a finite group, every element g satisfies gⁿ = e for some natural n ≥ 1 Lemma
• Integer addition and negation are well defined Lemma
• Integer multiplication is well defined Lemma
• Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
• Order is compatible with addition Lemma
• Order is compatible with multiplication Lemma
• The integer-defining relation is an equivalence relation Lemma
• The p-adic valuation extends to the nonzero rationals by vₚ(a/b) := vₚ(a) - vₚ(b) ∈ ℤ, independently of the representation; it satisfies vₚ(xy) = vₚ(x) + vₚ(y), and vₚ(x+y) ≥ min{vₚ(x), vₚ(y)} whenever x, y and x+y are nonzero Lemma
• Trichotomy of the order on ℕ Lemma
• ≤ is a linear order on ℕ Theorem
• Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b Theorem
• For all m and n there is a list of mn pairwise distinct reals with no strictly increasing sublist of length m+1 and no strictly decreasing sublist of length n+1 Theorem
• For every forest, |V|=|E|+c, where c is the number of connected components Theorem
• For m ≥ 1 the number of weak compositions of n into m parts is C(n+m-1, m-1), and the number of compositions is C(n-1, m-1) for n ≥ 1 Theorem
• For n ≥ 1 and any injective list p : r → ℤ of primes containing every prime divisor of n, one has n = ∏_i<r pᵢ^ v_pᵢ(n); the exponents are determined by n, and v_q(n) = 0 for every prime q outside the list Theorem
• For positive integers a and b and every prime p: vₚ(gcd(a,b)) = min{vₚ(a), vₚ(b)} and vₚ(lcm(a,b)) = max{vₚ(a), vₚ(b)}; so the exponent-wise greatest common divisor is the gcd of the divisibility page and not a second notion Theorem
• Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either Theorem
• ℕ × ℕ ≈ ℕ Theorem
• The arithmetic mean, geometric mean inequality Theorem
• The dimension formula: for finite-dimensional linear subspaces U and W of V, the subspaces U + W and U ∩ W are finite-dimensional and dim_F(U+W) + dim_F(U ∩ W) = dim_F U + dim_F W Theorem
• The integers form a commutative ring Theorem
• The integers form a totally ordered ring Theorem
• The Steinitz exchange lemma: if L ⊆ V is linearly independent and S ⊆ V spans V with S finite of size n, then L is finite with |L| = m ≤ n, and there is T ⊆ S of size n - m such that L ∪ T spans V Theorem
• The sum rule: a finite disjoint union is finite with | A ∪ B| = | A| + | B| and |⋃_i ∈ I Aᵢ| = ∑_i ∈ I| Aᵢ|, and a sum over a finite index set splits along a partition Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §2.1-2.3 (Peano axioms, recursion, arithmetic) (standard reference, not scraped)
• Mathematical induction (Wikipedia) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)
• B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)