Theorems · Theorem · ring theory

AddMonoidAlgebra.smul_eq

∀ {G : Type u_1} {P : Type u_2} {R : Type u_3} {V : Type u_4} [inst : VAdd G P] [inst_1 : IsLeftCancelVAdd G P]
  [inst_2 : Semiring R] [inst_3 : AddCommMonoid V] [inst_4 : SMulWithZero R V] (f : AddMonoidAlgebra R G) (x : P → V)
  (p : P) (hp : optParam ((↑f.coeff.support).vaddAntidiagonal (Function.support x) p).Finite ⋯),
  (f • x) p = ∑ gh ∈ Finset.VAddAntidiagonal p hp, f.coeff gh.1 • x gh.2

Defined in

Mathlib.Algebra.MonoidAlgebra.PointwiseSMul

Cited by

2 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

VAddIsLeftCancelVAddSemiringAddCommMonoidSMulWithZero