Theorems · Theorem · field theory
Polynomial.X_mul
∀ {R : Type u} [inst : Semiring R] {p : Polynomial R}, Polynomial.X * p = p * Polynomial.XX commutes with everything, even when the coefficients are noncommutative.
- Defined in
- Mathlib.Algebra.Polynomial.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 99 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.
Cites19
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
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- MulZeroClass.mul_zeroproof · cited by 2,091
- Polynomial.Xstatement · cited by 1,639
- MulZeroClass.zero_mulproof · cited by 1,625
- add_commproof · cited by 1,535
- AddMonoidAlgebraproof · cited by 649
- Finsupp.sumproof · cited by 481
- Finsupp.extproof · cited by 399
- AddMonoidAlgebra.coeffproof · cited by 365
Cited by12
Results whose statement or proof uses this declaration.
- Polynomial.commute_Xproof · cited by 13
- Polynomial.X_pow_mulproof · cited by 12
- Polynomial.X_mul_Cproof · cited by 9
- descPochhammer_eval_eq_ascPochhammerproof · cited by 4
- ascPochhammer_eval_zeroproof · cited by 3
- Polynomial.not_isFieldproof · cited by 3
- Polynomial.descPochhammer_smeval_eq_ascPochhammerproof · cited by 2
- eval_det_add_X_smulproof · cited by 2
- descPochhammer_eq_ascPochhammerproof · cited by 1
- Polynomial.X_mul_monomialproof · cited by 1
- descPochhammer_eval_zeroproof · cited by 1
- Polynomial.eval₂_X_mulproof · cited by 0