Prove2Me

`BookProof.ChapterA.cplxSub_isSubsystem` [CompleteSpace V] (M : System ℂ V) {Y : Submodule ℝ V} (hJ : ∀ y ∈ Y, (Complex.I : ℂ) • y ∈ Y) (hY : (rxSystem M).IsSubsystem Y) : (M).IsSu

Proved
BookProof.ChapterA.cplxSub_isSubsystem

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1d.

BookProof.ChapterA.cplxSub_isSubsystem [CompleteSpace V] (M : System ℂ V) {Y : Submodule ℝ V} (hJ : ∀ y ∈ Y, (Complex.I : ℂ) • y ∈ Y) (hY : (rxSystem M).IsSubsystem Y) : (M).IsSubsystem (cplxSub Y hJ)

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

Preamble
-- Generated from ChapterA1d.lean — theorem BookProof.ChapterA.cplxSub_isSubsystem
import Mathlib
import Definitions.Def_ChapterA1d
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace




attribute [local instance] InnerProductSpace.rclikeToReal

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
Formal statement
theorem BookProof.ChapterA.cplxSub_isSubsystem [CompleteSpace V] (M : System ℂ V) {Y : Submodule ℝ V}
    (hJ : ∀ y ∈ Y, (Complex.I : ℂ) • y ∈ Y) (hY : (rxSystem M).IsSubsystem Y) :
    (M).IsSubsystem (cplxSub Y hJ) := by sorry

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