`BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf` {T : D →ₗ[ℂ] F} {A : Dom →ₗ[ℂ] F} (hA : IsClosureOf T A) (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) : Dom = clDom T

Proved

BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterClosureUniqueness.

BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf {T : D →ₗ[ℂ] F} {A : Dom →ₗ[ℂ] F} (hA : IsClosureOf T A) (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) : Dom = clDom T ∧ ∀ (x : F) (h₁ : x ∈ Dom) (h₂ : x ∈ clDom T), A ⟨x, h₁⟩ = clExt T hdense hsym ⟨x, h₂⟩

Formalization note: Lean 4 identifier BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf.

Preamble

-- Generated from ChapterClosureUniqueness.lean — theorem BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterFarisLavineCore
open BookProof.EsaClosure
open BookProof.ClosureUniqueness



open BookProof.FarisLavine BookProof.EsaClosure

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable {D D₁ D₂ Dom Dom₁ Dom₂ : Submodule ℂ F}

Formal statement

theorem BookProof.ClosureUniqueness.eq_clExt_of_isClosureOf {T : D →ₗ[ℂ] F} {A : Dom →ₗ[ℂ] F} (hA : IsClosureOf T A)
    (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) :
    Dom = clDom T ∧ ∀ (x : F) (h₁ : x ∈ Dom) (h₂ : x ∈ clDom T),
      A ⟨x, h₁⟩ = clExt T hdense hsym ⟨x, h₂⟩ := by sorry

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