Theorems · Theorem · group theory

AddSubmonoid.apply_ne_zero_of_mem_or_neg_mem_closure

∀ {ι : Type u_1} {G : Type u_3} {S : Type u_4} [inst : AddCommGroup G] [inst_1 : LinearOrder S]
  [inst_2 : InvolutiveNeg ι] [inst_3 : AddCommGroup S] [IsOrderedAddMonoid S] (v : ι → G) (f : G →+ S) (s : Set ι),
  (∀ i ∈ s, 0 < f (v i)) →
    ∀ (i : ι),
      v i ≠ 0 → v (-i) = -v i → v i ∈ AddSubmonoid.closure (v '' s) ∨ -v i ∈ AddSubmonoid.closure (v '' s) → f (v i) ≠ 0

Defined in

Mathlib.Algebra.Group.Irreducible.Indecomposable

Cited by

3 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupLinearOrderInvolutiveNegAddCommGroupIsOrderedAddMonoid