Theorems · Theorem · nonassociative algebras

RootPairing.EmbeddedG2.mem_span_of_mem_allRoots

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (P : RootPairing ι R M N)
  [inst_5 : P.EmbeddedG2] [Finite ι] [CharZero R] [IsDomain R] {x : M},
  x ∈ RootPairing.EmbeddedG2.allRoots P →
    x ∈ Submodule.span ℤ {RootPairing.EmbeddedG2.longRoot P, RootPairing.EmbeddedG2.shortRoot P}

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.G2

Cited by

1 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleRootPairing.EmbeddedG2FiniteCharZeroIsDomain