`BookProof.DirectSumEsa.dsOpD_coe` (A : ∀ i, D i →ₗ[ℂ] D i) (x : dsCore D) : ((dsOpD A x : dsCore D) : lp G 2) = (dsOp (fun i => (D i).subtype.comp (A i)) x : lp G 2)

Proved

BookProof.DirectSumEsa.dsOpD_coe

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDirectSumEsa.

BookProof.DirectSumEsa.dsOpD_coe (A : ∀ i, D i →ₗ[ℂ] D i) (x : dsCore D) : ((dsOpD A x : dsCore D) : lp G 2) = (dsOp (fun i => (D i).subtype.comp (A i)) x : lp G 2)

Formalization note: Lean 4 identifier BookProof.DirectSumEsa.dsOpD_coe.

Preamble

-- Generated from ChapterDirectSumEsa.lean — theorem BookProof.DirectSumEsa.dsOpD_coe
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterDirectSumEsa
open BookProof.DirectSumEsa


open scoped ENNReal


open BookProof.FarisLavine

noncomputable section

variable {ι : Type*} {G : ι → Type*} [∀ i, NormedAddCommGroup (G i)]
  [∀ i, InnerProductSpace ℂ (G i)]

variable {D : ∀ i, Submodule ℂ (G i)}
variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]

Formal statement

theorem BookProof.DirectSumEsa.dsOpD_coe (A : ∀ i, D i →ₗ[ℂ] D i) (x : dsCore D) :
    ((dsOpD A x : dsCore D) : lp G 2)
      = (dsOp (fun i => (D i).subtype.comp (A i)) x : lp G 2) := by sorry

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