Theorems · Theorem · several complex variables

FormalMultilinearSeries.removeZero_comp_of_pos

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : CommRing 𝕜] [inst_1 : AddCommGroup E]
  [inst_2 : AddCommGroup F] [inst_3 : AddCommGroup G] [inst_4 : Module 𝕜 E] [inst_5 : Module 𝕜 F] [inst_6 : Module 𝕜 G]
  [inst_7 : TopologicalSpace E] [inst_8 : TopologicalSpace F] [inst_9 : TopologicalSpace G]
  [inst_10 : IsTopologicalAddGroup E] [inst_11 : ContinuousConstSMul 𝕜 E] [inst_12 : IsTopologicalAddGroup F]
  [inst_13 : ContinuousConstSMul 𝕜 F] [inst_14 : IsTopologicalAddGroup G] [inst_15 : ContinuousConstSMul 𝕜 G]
  (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ},
  0 < n → q.removeZero.comp p n = q.comp p n

Only 0-th coefficient of q.comp p depends on q 0.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupAddCommGroupAddCommGroupModuleModuleModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceIsTopologicalAddGroupContinuousConstSMulIsTopologicalAddGroupContinuousConstSMulIsTopologicalAddGroupContinuousConstSMul