Multiplication is commutative

Depends on

• Multiplication of natural numbers
• Zero and one under multiplication
• Distributivity and the successor law for multiplication
• The principle of mathematical induction

Used by

• ⌈ m/n ⌉ for naturals m and n ≥ 1: the least q ∈ ℕ with m ≤ n q Definition
• Exponentiation of natural numbers, mⁿ, and its agreement with the integer power in ℝ Definition
• The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ Definition
• FALSE: for all sets A and B with B having at least two elements, A × B is strictly larger than A False statement
• 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
• 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 multiplication Lemma
• A finite set with n elements has exactly C(n, 2) two-element subsets, and 2C(n, 2) = n(n-1) Theorem
• C(n, k) k! (n-k)! = n! for k ≤ n; hence C(n, k) k! = n^k̲, the quotient n!/(k!(n-k)!) is a natural number, and C(n, k) = C(n, n-k) 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
• If | A| > k| B| then every f : A → B has a fibre with more than k elements, and for nonempty B some fibre has at least ⌈ | A| / | B|⌉ elements Theorem
• ℕ × ℕ ≈ ℕ Theorem
• The integers form a commutative ring Theorem
• The integers form a totally ordered ring Theorem
• The multinomial coefficient equals n!/∏_i<m kᵢ!, and (x₀+…+xₘ₋₁)ⁿ = ∑ iotaC(n, k)∏_i<m xᵢ^kᵢ in ℝ Theorem
• The product rule: | A × B| = | A| | B|, and |∏_i<m Aᵢ| = ∏_i<m| Aᵢ| Theorem

Dependency tree · two levels

13 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)
• Peano axioms (Wikipedia) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)