Addition is associative

Depends on

• Addition of natural numbers
• 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
• 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
• Distributivity and the successor law for multiplication Lemma
• Finite words satisfy the free-monoid universal property 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
• The integer-defining relation is an equivalence relation Lemma
• Trichotomy of the order on ℕ Lemma
• ≤ is a linear order on ℕ Theorem
• For every forest, |V|=|E|+c, where c is the number of connected components 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

Dependency tree · two levels

7 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)
• Mathematics 144: Set Theory (UC Riverside lecture notes) (standard reference, not scraped)