`BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply` (κ : J → ℝ) (x : lpFiniteModes (ℕ × J)) (j : J) (n : ℕ) : ((bilH κ x : L2I (ℕ × J)) : ℕ × J → ℂ) (n, j) = hFun...

Proved

BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNavierStokesBilinearEsa.

BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply (κ : J → ℝ) (x : lpFiniteModes (ℕ × J)) (j : J) (n : ℕ) : ((bilH κ x : L2I (ℕ × J)) : ℕ × J → ℂ) (n, j) = hFun (κ j) (fun m => ((x : L2I (ℕ × J)) : ℕ × J → ℂ) (m, j)) n

Formalization note: Lean 4 identifier BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply.

Preamble

-- Generated from ChapterNavierStokesBilinearEsa.lean — theorem BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply
import Mathlib
import Definitions.Def_ChapterNavierStokesBilinearEsa
import Definitions.Def_ChapterNavierStokesEsa
import Definitions.Def_ChapterNavierStokesHermiteFarisLavine
import Definitions.Def_ChapterNavierStokesIkebeKato
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.BilinearEsa


open scoped ENNReal



open LpNat BookProof.FarisLavine IkebeKato HermiteFarisLavine

variable {J : Type*}

Formal statement

theorem BookProof.NavierStokesFlow.BilinearEsa.blockVec_bilH_apply (κ : J → ℝ) (x : lpFiniteModes (ℕ × J)) (j : J) (n : ℕ) :
    ((bilH κ x : L2I (ℕ × J)) : ℕ × J → ℂ) (n, j)
      = hFun (κ j) (fun m => ((x : L2I (ℕ × J)) : ℕ × J → ℂ) (m, j)) n := by sorry

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