Theorems · Theorem · nonassociative algebras

LieSubalgebra.lieSpan_induction

∀ (R : Type u) {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] {s : Set L}
  {p : (x : L) → x ∈ LieSubalgebra.lieSpan R L s → Prop},
  (∀ (x : L) (h : x ∈ s), p x ⋯) →
    p 0 ⋯ →
      (∀ (x y : L) (hx : x ∈ LieSubalgebra.lieSpan R L s) (hy : y ∈ LieSubalgebra.lieSpan R L s),
          p x hx → p y hy → p (x + y) ⋯) →
        (∀ (a : R) (x : L) (hx : x ∈ LieSubalgebra.lieSpan R L s), p x hx → p (a • x) ⋯) →
          ∀ {x : L},
            (∀ (x y : L) (hx : x ∈ LieSubalgebra.lieSpan R L s) (hy : y ∈ LieSubalgebra.lieSpan R L s),
                p x hx → p y hy → p ⁅x, y⁆ ⋯) →
              ∀ (hx : x ∈ LieSubalgebra.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 algebra spanned by s.

Defined in

Mathlib.Algebra.Lie.Subalgebra

Cited by

9 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebra