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Mathlib.SetTheory.Cardinal.Divisibility

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 #

theorem Cardinal.dvd_of_le_of_aleph0_le {a b : Cardinal.{u}} (ha : a 0) (h : a b) (hb : aleph0 b) :
@[simp]
@[simp]