Theorems · Theorem · field theory
Polynomial.natDegree_sub_eq_of_prod_eq
∀ {R : Type u} [inst : Semiring R] [NoZeroDivisors R] {p₁ p₂ q₁ q₂ : Polynomial R},
p₁ ≠ 0 → q₁ ≠ 0 → p₂ ≠ 0 → q₂ ≠ 0 → p₁ * q₂ = p₂ * q₁ → ↑p₁.natDegree - ↑q₁.natDegree = ↑p₂.natDegree - ↑q₂.natDegreeThis lemma is useful for working with the intDegree of a rational function.
- Defined in
- Mathlib.Algebra.Polynomial.Degree.Domain
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Polynomialstatement and proof · cited by 5,681
- Polynomial.natDegreestatement and proof · cited by 1,105
- NoZeroDivisorsstatement and proof · cited by 545
- Polynomial.natDegree_mulproof · cited by 23
- sub_eq_sub_iff_add_eq_addproof · cited by 9
Cited by3
Results whose statement or proof uses this declaration.
- RatFunc.intDegree_addproof · cited by 1
- RatFunc.intDegree_negproof · cited by 0