Prove2Me

`BookProof.ChapterA.mem_JY` {Y : Submodule ℝ V} {x : V} : x ∈ JY Y ↔ ∃ y ∈ Y, Jmap y = x

Proved
BookProof.ChapterA.mem_JY

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1e.

BookProof.ChapterA.mem_JY {Y : Submodule ℝ V} {x : V} : x ∈ JY Y ↔ ∃ y ∈ Y, Jmap y = x

Formalization note: Lean 4 identifier BookProof.ChapterA.mem_JY.

Preamble
-- Generated from ChapterA1e.lean — theorem BookProof.ChapterA.mem_JY
import Mathlib
import Definitions.Def_ChapterA1e
import Definitions.Def_ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace


attribute [local instance] InnerProductSpace.rclikeToReal

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
Formal statement
theorem BookProof.ChapterA.mem_JY {Y : Submodule ℝ V} {x : V} : x ∈ JY Y ↔ ∃ y ∈ Y, Jmap y = x := by sorry

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