Prove2Me

`BookProof.ChapterA.realSub_isSubsystem` [CompleteSpace V] (M : System ℂ V) {X : Submodule ℂ V} (hX : (M).IsSubsystem X) : (rxSystem M).IsSubsystem (realSub X)

Proved
BookProof.ChapterA.realSub_isSubsystem

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1d.

BookProof.ChapterA.realSub_isSubsystem [CompleteSpace V] (M : System ℂ V) {X : Submodule ℂ V} (hX : (M).IsSubsystem X) : (rxSystem M).IsSubsystem (realSub X)

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

Preamble
-- Generated from ChapterA1d.lean — theorem BookProof.ChapterA.realSub_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.realSub_isSubsystem [CompleteSpace V] (M : System ℂ V) {X : Submodule ℂ V}
    (hX : (M).IsSubsystem X) : (rxSystem M).IsSubsystem (realSub X) := by sorry

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