Theorems · Theorem · commutative algebra

PolynomialModule.smul_single_apply

∀ {R : Type u_2} {M : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (i : ℕ)
  (f : Polynomial R) (m : M) (n : ℕ),
  (f • PolynomialModule.single R i m).coeff n = if i ≤ n then f.coeff (n - i) • m else 0

Defined in: Mathlib.Algebra.Polynomial.Module.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module

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Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
Polynomialstatement and proof · cited by 5,681
Finsuppstatement · cited by 5,255
add_zeroproof · cited by 2,707
Polynomial.coeffstatement and proof · cited by 1,045
Polynomial.monomialproof · cited by 256
add_smulproof · cited by 204
Polynomial.coeff_addproof · cited by 77
PolynomialModulestatement · cited by 76

Cited by1

Results whose statement or proof uses this declaration.

Ideal.Filtration.submodule_eq_span_le_iff_stable_geproof · cited by 1