Theorems · Theorem · ring theory

listProd_apply_eq_zero

∀ {ι : Type u_1} {σ : Type u_2} {R : Type u_4} [inst : AddCommMonoid ι] [inst_1 : LinearOrder ι] [IsOrderedAddMonoid ι]
  [inst_3 : DecidableEq ι] [inst_4 : Semiring R] [inst_5 : SetLike σ R] [inst_6 : AddSubmonoidClass σ R] {A : ι → σ}
  [inst_7 : SetLike.GradedMonoid A] [CanonicallyOrderedAdd ι] {l : List (DirectSum ι fun i => ↥(A i))} {m : ι},
  (∀ x ∈ l, ∀ k < m, x k = 0) → ∀ ⦃n : ι⦄, n < l.length • m → l.prod n = 0

Defined in

Mathlib.Algebra.DirectSum.Internal

Cited by

2 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidLinearOrderIsOrderedAddMonoidDecidableEqSemiringSetLikeAddSubmonoidClassSetLike.GradedMonoidCanonicallyOrderedAdd