Theorems · Theorem · field theory
Polynomial.natDegree_mul
∀ {R : Type u} [inst : Semiring R] [NoZeroDivisors R] {p q : Polynomial R},
p ≠ 0 → q ≠ 0 → (p * q).natDegree = p.natDegree + q.natDegree- Defined in
- Mathlib.Algebra.Polynomial.Degree.Domain
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 91 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.
Cites11
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
- WithBotproof · cited by 1,498
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.degreeproof · cited by 643
- Nat.cast_addproof · cited by 586
- NoZeroDivisorsstatement and proof · cited by 545
- mul_ne_zeroproof · cited by 178
- Nat.cast_injproof · cited by 70
- Polynomial.degree_eq_natDegreeproof · cited by 69
- Polynomial.degree_mulproof · cited by 25
Cited by23
Results whose statement or proof uses this declaration.
- Polynomial.natDegree_le_of_dvdproof · cited by 11
- Polynomial.associated_content_mulproof · cited by 6
- Polynomial.natDegree_eq_zero_of_isUnitproof · cited by 5
- Polynomial.splits_mulproof · cited by 4
- Polynomial.natDegree_sub_eq_of_prod_eqproof · cited by 3
- AdjoinRoot.minpoly_rootproof · cited by 3
- RatFunc.intDegree_mulproof · cited by 3
- MvPolynomial.degreeOf_mul_eqproof · cited by 2
- Polynomial.cyclotomic_expand_eq_cyclotomic_mulproof · cited by 2
- Polynomial.natDegree_mul_mirrorproof · cited by 2
- Irreducible.natDegree_le_twoproof · cited by 2
- ascPochhammer_natDegreeproof · cited by 2