Theorems · Theorem · global analysis

fderivWithin_const_smul_field

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {x : E} {s : Set E} {R : Type u_4} [inst_5 : DivisionSemiring R] [inst_6 : Module R F] [inst_7 : SMulCommClass 𝕜 R F]
  [inst_8 : ContinuousConstSMul R F] (c : R),
  UniqueDiffWithinAt 𝕜 s x → fderivWithin 𝕜 (c • f) s x = c • fderivWithin 𝕜 f s x

Special case of fderivWithin_const_smul_of_invertible over a division semiring: any constant is allowed. TODO: This would work for scalars in a GroupWithZero if we had a DistribMulActionWithZero typeclass.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Add

Cited by

6 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceDivisionSemiringModuleSMulCommClassContinuousConstSMul