`BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self` {D : Submodule ℂ F} {A : D →ₗ[ℂ] D} (hA : IsSymmetricDom A) (v : D) : (inner ℂ (v : F) ((A.comp A) v :

Proved

BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNavierStokesLagrangianEsa.

BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self {D : Submodule ℂ F} {A : D →ₗ[ℂ] D} (hA : IsSymmetricDom A) (v : D) : (inner ℂ (v : F) ((A.comp A) v : F) : ℂ) = ((‖(A v : F)‖ ^ 2 : ℝ) : ℂ)

Formalization note: Lean 4 identifier BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self.

Preamble

-- Generated from ChapterNavierStokesLagrangianEsa.lean — theorem BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self
import Mathlib
import Definitions.Def_ChapterNavierStokesLagrangianEsa
import Definitions.Def_ChapterNavierStokesFullEsa
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.FullEsa.NSFullData
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.LagrangianEsa




open FullEsa

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable (L : LagrangianFullData F)

Formal statement

theorem BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.inner_comp_self {D : Submodule ℂ F} {A : D →ₗ[ℂ] D} (hA : IsSymmetricDom A) (v : D) :
    (inner ℂ (v : F) ((A.comp A) v : F) : ℂ) = ((‖(A v : F)‖ ^ 2 : ℝ) : ℂ) := by sorry

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