`BookProof.ChapterCayleyInverse.invCayleyOp_cayley` (x : T.domain) (hmem : (x : H) ∈ invCayleyDomain (cayley T)) : invCayleyOp (cayley T) (oneSubU_cayley_injective T) ⟨(x : H), hme

Proved

BookProof.ChapterCayleyInverse.invCayleyOp_cayley

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCayleyInverse.

BookProof.ChapterCayleyInverse.invCayleyOp_cayley (x : T.domain) (hmem : (x : H) ∈ invCayleyDomain (cayley T)) : invCayleyOp (cayley T) (oneSubU_cayley_injective T) ⟨(x : H), hmem⟩ = T.op x

Formalization note: Lean 4 identifier BookProof.ChapterCayleyInverse.invCayleyOp_cayley.

Preamble

-- Generated from ChapterCayleyInverse.lean — theorem BookProof.ChapterCayleyInverse.invCayleyOp_cayley
import Definitions.Def_ChapterUnitaryTransport
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterCayleyInverse
import Definitions.Def_ChapterCayleyTransform
import Definitions.Def_ChapterStoneResolvent
import Theorems.Thm_BookProof_ChapterCayleyInverse_oneSubU_cayley_injective
open BookProof.ChapterCayleyTransform
open BookProof.ChapterCayleyInverse


open scoped InnerProductSpace


open BookProof.ChapterUnitaryTransport BookProof.ChapterStoneResolvent
open BookProof.ChapterCayleyTransform

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

variable (V : H ≃ₗᵢ[ℂ] H)
variable (hinj : Function.Injective (oneSubU V))
variable [CompleteSpace H]
variable (T : UnboundedSelfAdjoint H)

Formal statement

theorem BookProof.ChapterCayleyInverse.invCayleyOp_cayley (x : T.domain) (hmem : (x : H) ∈ invCayleyDomain (cayley T)) :
    invCayleyOp (cayley T) (oneSubU_cayley_injective T) ⟨(x : H), hmem⟩ = T.op x := by sorry

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