Theorems · Theorem · field theory
Polynomial.natDegree_mul_le
∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R}, (p * q).natDegree ≤ p.natDegree + q.natDegree- Defined in
- Mathlib.Algebra.Polynomial.Degree.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
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
- le_transproof · cited by 985
- add_le_addproof · cited by 666
- Polynomial.degreeproof · cited by 643
- Nat.cast_addproof · cited by 586
- Polynomial.natDegree_le_of_degree_leproof · cited by 16
- Polynomial.degree_le_natDegreeproof · cited by 14
- Polynomial.degree_mul_leproof · cited by 10
Cited by15
Results whose statement or proof uses this declaration.
- Polynomial.natDegree_C_mul_leproof · cited by 8
- Polynomial.natDegree_mul_le_of_leproof · cited by 8
- Polynomial.coeff_comp_degree_mul_degreeproof · cited by 7
- Polynomial.natDegree_pow_leproof · cited by 6
- IsIntegral.coeffproof · cited by 4
- Polynomial.natDegree_mul_C_leproof · cited by 4
- Polynomial.isUnit_resultant_iff_isCoprimeproof · cited by 2
- Polynomial.mul_scaleRootsproof · cited by 2
- Polynomial.Splits.natDegree_le_one_of_irreducibleproof · cited by 2
- Polynomial.natDegree_list_prod_leproof · cited by 2
- isIntegral_of_isIntegral_adjoin_of_mul_eq_oneproof · cited by 1
- Polynomial.IsWeaklyEisensteinAt.mulproof · cited by 1