Theorems · Theorem · global analysis

mvfderivWithin_smul

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_8} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {s : Set M} {x : M} {a : M → 𝕜},
  MDiffAt[s] a x →
    ∀ {g : M → F}, MDiffAt[s] g x → UniqueMDiffAt[s] x → d[s] (a • g) x = a x • d[s] g x + (d[s] a x).smulRight (g x)

Defined in

Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpace