`BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset` (A : E →L[ℂ] E) {ω : ℝ} : NumReLE A ω ↔ BookProof.ChapterH9.numRange A ⊆ {z : ℂ | z.re ≤ ω}

Proved

BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeSemigroup.

BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset (A : E →L[ℂ] E) {ω : ℝ} : NumReLE A ω ↔ BookProof.ChapterH9.numRange A ⊆ {z : ℂ | z.re ≤ ω}

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset.

Preamble

-- Generated from ChapterNumericalRangeSemigroup.lean — theorem BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset
import Mathlib
import Definitions.Def_ChapterNumericalRangeSemigroup
import Definitions.Def_ChapterH9
open BookProof.ChapterH9
open BookProof.ChapterNumericalRangeSemigroup


open scoped InnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]

Formal statement

theorem BookProof.ChapterNumericalRangeSemigroup.numReLE_iff_numRange_subset (A : E →L[ℂ] E) {ω : ℝ} :
    NumReLE A ω ↔ BookProof.ChapterH9.numRange A ⊆ {z : ℂ | z.re ≤ ω} := by sorry

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