`BookProof.ReducedEsa.symmetric_of_involutive_isometry` (hU2 : ∀ x, U (U x) = x) (hUi : ∀ x y : F, (inner ℂ (U x) (U y) : ℂ) = inner ℂ x y) (x y : F) : (inner ℂ (U x) y : ℂ) = inne

Proved

BookProof.ReducedEsa.symmetric_of_involutive_isometry

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterReducingSubspaceEsa.

BookProof.ReducedEsa.symmetric_of_involutive_isometry (hU2 : ∀ x, U (U x) = x) (hUi : ∀ x y : F, (inner ℂ (U x) (U y) : ℂ) = inner ℂ x y) (x y : F) : (inner ℂ (U x) y : ℂ) = inner ℂ x (U y)

Formalization note: Lean 4 identifier BookProof.ReducedEsa.symmetric_of_involutive_isometry.

Preamble

-- Generated from ChapterReducingSubspaceEsa.lean — theorem BookProof.ReducedEsa.symmetric_of_involutive_isometry
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Mathlib
import Definitions.Def_ChapterReducingSubspaceEsa
open BookProof.ReducedEsa



open BookProof.FarisLavine BookProof.GraphCore

noncomputable section

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

variable {P : F →ₗ[ℂ] F}
variable {D : Submodule ℂ F}
variable (T : D →ₗ[ℂ] F)
variable {T}
variable (U : F →ₗ[ℂ] F)
variable {U}

Formal statement

theorem BookProof.ReducedEsa.symmetric_of_involutive_isometry (hU2 : ∀ x, U (U x) = x)
    (hUi : ∀ x y : F, (inner ℂ (U x) (U y) : ℂ) = inner ℂ x y) (x y : F) :
    (inner ℂ (U x) y : ℂ) = inner ℂ x (U y) := by sorry

View graph