Theorems · Theorem · commutative algebra

mul_eq_mul_prime_pow

∀ {R : Type u_1} [inst : CommMonoidWithZero R] [IsCancelMulZero R] {x y a p : R} {n : ℕ},
  Prime p → x * y = a * p ^ n → ∃ i j b c, i + j = n ∧ a = b * c ∧ x = b * p ^ i ∧ y = c * p ^ j

If x * y = a * p ^ n where p is prime, then x and y can both be written as the product of a power of p and a divisor of a.

Defined in

Mathlib.RingTheory.Prime

Cited by

0 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidWithZeroIsCancelMulZero