Theorems · Theorem · nonassociative algebras

LieSubmodule.lieSpan_induction

∀ (R : Type u) (L : Type v) {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : AddCommGroup M]
  [inst_3 : Module R M] [inst_4 : LieRingModule L M] {s : Set M} {p : (x : M) → x ∈ LieSubmodule.lieSpan R L s → Prop},
  (∀ (x : M) (h : x ∈ s), p x ⋯) →
    p 0 ⋯ →
      (∀ (x y : M) (hx : x ∈ LieSubmodule.lieSpan R L s) (hy : y ∈ LieSubmodule.lieSpan R L s),
          p x hx → p y hy → p (x + y) ⋯) →
        (∀ (a : R) (x : M) (hx : x ∈ LieSubmodule.lieSpan R L s), p x hx → p (a • x) ⋯) →
          ∀ {x : M},
            (∀ (x : L) (y : M) (hy : y ∈ LieSubmodule.lieSpan R L s), p y hy → p ⁅x, y⁆ ⋯) →
              ∀ (hx : x ∈ LieSubmodule.lieSpan R L s), p x hx

An induction principle for span membership. If p holds for 0 and all elements of s, and is preserved under addition, scalar multiplication and the Lie bracket, then p holds for all elements of the Lie submodule spanned by s.

Defined in

Mathlib.Algebra.Lie.Submodule

Cited by

0 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

CommRingLieRingAddCommGroupModuleLieRingModule