Theorems · Theorem · global analysis

VectorField.leibniz_identity_lieBracket

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E],
  minSmoothness 𝕜 2 ≤ n →
    ∀ {U V W : E → E} {x : E},
      ContDiffAt 𝕜 n U x →
        ContDiffAt 𝕜 n V x →
          ContDiffAt 𝕜 n W x →
            VectorField.lieBracket 𝕜 U (VectorField.lieBracket 𝕜 V W) x =
              VectorField.lieBracket 𝕜 (VectorField.lieBracket 𝕜 U V) W x +
                VectorField.lieBracket 𝕜 V (VectorField.lieBracket 𝕜 U W) x

The Lie bracket of vector fields in vector spaces satisfies the Leibniz identity [U, [V, W]] = [[U, V], W] + [V, [U, W]].

Defined in: Mathlib.Analysis.Calculus.VectorField
Cited by: 0 results in Mathlib
Foundations: Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField NormedAddCommGroup NormedSpace

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
NontriviallyNormedFieldstatement and proof · cited by 8,742
ENatstatement and proof · cited by 4,985
WithTopstatement and proof · cited by 3,754
closureproof · cited by 1,254
Set.mem_univproof · cited by 416
ContDiffAtstatement and proof · cited by 262
IsClosed.closure_eqproof · cited by 139
uniqueDiffOn_univproof · cited by 66
minSmoothnessstatement and proof · cited by 50
VectorField.lieBracketstatement and proof · cited by 27

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