`BookProof.ChapterA.IsRReal.excludes_other_types` [Nontrivial W] {M : System ℝ W} (h : IsRReal M) : IsRRealType M ∧ ¬ IsRComplexType M ∧ ¬ IsRPseudorealType M

Proved

BookProof.ChapterA.IsRReal.excludes_other_types

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1h.

BookProof.ChapterA.IsRReal.excludes_other_types [Nontrivial W] {M : System ℝ W} (h : IsRReal M) : IsRRealType M ∧ ¬ IsRComplexType M ∧ ¬ IsRPseudorealType M

Formalization note: Lean 4 identifier BookProof.ChapterA.IsRReal.excludes_other_types.

Preamble

-- Generated from ChapterA1h.lean — theorem BookProof.ChapterA.IsRReal.excludes_other_types
import Definitions.Def_Complexification
import Mathlib
import Definitions.Def_ChapterA1h
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace


open BookProof.Complexification

variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [CompleteSpace W]

Formal statement

theorem BookProof.ChapterA.IsRReal.excludes_other_types [Nontrivial W] {M : System ℝ W}
    (h : IsRReal M) : IsRRealType M ∧ ¬ IsRComplexType M ∧ ¬ IsRPseudorealType M := by sorry

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