# mathlibdocumentation

set_theory.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_aleph_0_le), showing that the cardinal numbers do not form a comm_cancel_monoid_with_zero.

## Main results #

• cardinal.prime_of_aleph_0_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.is_prime_pow_iff: a cardinal is a prime power iff it is infinite or a natural number which is itself a prime power.
@[simp]
theorem cardinal.is_unit_iff {a : cardinal} :
a = 1
@[protected, instance]
Equations
theorem cardinal.le_of_dvd {a b : cardinal} :
b 0a ba b
theorem cardinal.dvd_of_le_of_aleph_0_le {a b : cardinal} (ha : a 0) (h : a b) (hb : cardinal.aleph_0 b) :
a b
@[simp, norm_cast]
theorem cardinal.nat_coe_dvd_iff {n m : } :
n m n m
@[simp]
theorem cardinal.nat_is_prime_iff {n : } :
theorem cardinal.is_prime_iff {a : cardinal} :
∃ (p : ), a = p
theorem cardinal.is_prime_pow_iff {a : cardinal} :
∃ (n : ), a = n