Cardinal Divisibility

We show basic results about divisibility in the cardinal numbers. This relation can be characterised in the following simple way: if a and b are both less than ℵ₀, then a ∣ b iff they are divisible as natural numbers. If b is greater than ℵ₀, then a ∣ b iff a ≤ b. This furthermore shows that all infinite cardinals are prime; recall that a * b = max a b if ℵ₀ ≤ a * b; therefore a ∣ b * c = a ∣ max b c and therefore clearly either a ∣ b or a ∣ c. Note furthermore that no infinite cardinal is irreducible (Cardinal.not_irreducible_of_aleph0_le), showing that the cardinal numbers do not form a CancelCommMonoidWithZero.

Main results

• Cardinal.prime_of_aleph0_le: a Cardinal is prime if it is infinite.
• Cardinal.is_prime_iff: a Cardinal is prime iff it is infinite or a prime natural number.
• Cardinal.isPrimePow_iff: a Cardinal is a prime power iff it is infinite or a natural number which is itself a prime power.

theorem Cardinal.isUnit_iff {a : Cardinal.{u}} : IsUnit a ↔ a = 1

Alias of isUnit_iff_eq_one for discoverability.

instance Cardinal.instUniqueUnits : Unique Cardinal.{u}ˣ

Equations

• Cardinal.instUniqueUnits = { default := 1, uniq := Cardinal.instUniqueUnits._proof_1 }

theorem Cardinal.le_of_dvd {a b : Cardinal.{u_1}} : b ≠ 0 → a ∣ b → a ≤ b

theorem Cardinal.dvd_of_le_of_aleph0_le {a b : Cardinal.{u}} (ha : a ≠ 0) (h : a ≤ b) (hb : aleph0 ≤ b) : a ∣ b

@[simp]

theorem Cardinal.prime_of_aleph0_le {a : Cardinal.{u}} (ha : aleph0 ≤ a) : Prime a

theorem Cardinal.not_irreducible_of_aleph0_le {a : Cardinal.{u}} (ha : aleph0 ≤ a) : ¬Irreducible a

@[simp]

theorem Cardinal.nat_coe_dvd_iff {n m : ℕ} : ↑n ∣ ↑m ↔ n ∣ m

@[simp]

theorem Cardinal.nat_is_prime_iff {n : ℕ} : Prime ↑n ↔ Nat.Prime n

theorem Cardinal.is_prime_iff {a : Cardinal.{u_1}} : Prime a ↔ aleph0 ≤ a ∨ ∃ (p : ℕ), a = ↑p ∧ Nat.Prime p

theorem Cardinal.isPrimePow_iff {a : Cardinal.{u_1}} : IsPrimePow a ↔ aleph0 ≤ a ∨ ∃ (n : ℕ), a = ↑n ∧ IsPrimePow n