`BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe` (x : maxDom P.sym) (β : ι) : ((pairH P x : L2I ι) : ι → ℂ) β = P.fst.hFun ((x : L2I ι) : ι → ℂ) β + P.snd.hFun ((x : L2

Proved

BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNavierStokesAffineFiberEsa.

BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe (x : maxDom P.sym) (β : ι) : ((pairH P x : L2I ι) : ι → ℂ) β = P.fst.hFun ((x : L2I ι) : ι → ℂ) β + P.snd.hFun ((x : L2I ι) : ι → ℂ) β

Formalization note: Lean 4 identifier BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe.

Preamble

-- Generated from ChapterNavierStokesAffineFiberEsa.lean — theorem BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe
import Mathlib
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Definitions.Def_ChapterNavierStokesHermiteFarisLavine
import Definitions.Def_ChapterNavierStokesIkebeKato
import Definitions.Def_ChapterNavierStokesShiftHamiltonian
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.ShiftHamiltonian.ShiftData
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.AffineFiber


open scoped ENNReal



open LpNat BookProof.FarisLavine IkebeKato HermiteFarisLavine ShiftHamiltonian

variable {ι : Type*}

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] {D : Submodule ℂ F}
variable (P : PairShift ι)

Formal statement

theorem BookProof.NavierStokesFlow.AffineFiber.PairShift.pairH_coe (x : maxDom P.sym) (β : ι) :
    ((pairH P x : L2I ι) : ι → ℂ) β
      = P.fst.hFun ((x : L2I ι) : ι → ℂ) β + P.snd.hFun ((x : L2I ι) : ι → ℂ) β := by sorry

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