Theorems · Theorem · complex analysis

fun_meromorphicOrderAt_smul

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {R : Type u_3} [inst_3 : NormedRing R] [NoZeroDivisors R] [inst_5 : Module R E]
  [IsBoundedSMul R E] [Module.IsTorsionFree R E] {x : 𝕜} [inst_8 : NormedAlgebra 𝕜 R] [IsScalarTower 𝕜 R E] {f : 𝕜 → R}
  {g : 𝕜 → E},
  MeromorphicAt f x →
    MeromorphicAt g x → meromorphicOrderAt (fun i => f i • g i) x = meromorphicOrderAt f x + meromorphicOrderAt g x

Eta-expanded form of meromorphicOrderAt_smul The order is additive when multiplying scalar-valued and vector-valued meromorphic functions.

Defined in

Mathlib.Analysis.Meromorphic.Order

Cited by

0 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedRingNoZeroDivisorsModuleIsBoundedSMulModule.IsTorsionFreeNormedAlgebraIsScalarTower