`BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary` (W : H ≃ₗᵢ[ℂ] K) (D : Submodule ℂ H) (A : D →ₗ[ℂ] H) (U : ℝ → H ≃ₗᵢ[ℂ] H) (h : ∀ x : D,...

Proved

BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterUnitaryTransport.

BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary (W : H ≃ₗᵢ[ℂ] K) (D : Submodule ℂ H) (A : D →ₗ[ℂ] H) (U : ℝ → H ≃ₗᵢ[ℂ] H) (h : ∀ x : D, Filter.Tendsto (fun t : ℝ => (t⁻¹ : ℝ) • (U t (x : H) - (x : H))) (nhdsWithin 0 {0}ᶜ) (nhds (Complex.I • A x))) (y : transportDomain W D) : Filter.Tendsto (fun t : ℝ => (t⁻¹ : ℝ) • (transportUnitary W (U t) (y : K) - (y : K))) (nhdsWithin 0 {0}ᶜ) (nhds (Complex.I • transportOp W D A y))

Formalization note: Lean 4 identifier BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary.

Preamble

-- Generated from ChapterUnitaryTransport.lean — theorem BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary
import Mathlib
import Definitions.Def_ChapterUnitaryTransport
open BookProof.ChapterUnitaryTransport


open scoped InnerProductSpace


variable {H K : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
  [NormedAddCommGroup K] [InnerProductSpace ℂ K]

Formal statement

theorem BookProof.ChapterUnitaryTransport.tendsto_slope_transportUnitary (W : H ≃ₗᵢ[ℂ] K) (D : Submodule ℂ H) (A : D →ₗ[ℂ] H)
    (U : ℝ → H ≃ₗᵢ[ℂ] H)
    (h : ∀ x : D, Filter.Tendsto (fun t : ℝ => (t⁻¹ : ℝ) • (U t (x : H) - (x : H)))
      (nhdsWithin 0 {0}ᶜ) (nhds (Complex.I • A x)))
    (y : transportDomain W D) :
    Filter.Tendsto (fun t : ℝ => (t⁻¹ : ℝ) • (transportUnitary W (U t) (y : K) - (y : K)))
      (nhdsWithin 0 {0}ᶜ) (nhds (Complex.I • transportOp W D A y)) := by sorry

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