`BookProof.NonnegSemigroup.norm_semigroupS_apply_le` (hT : IsNonnegSelfAdjoint T) (hsv : ∀ w : F, ((0 : F), w) ∈ T → w = 0) {t : ℝ} (ht : 0 ≤ t) (x : F) : ‖semigroupS hT hsv ht x‖

Proved

BookProof.NonnegSemigroup.norm_semigroupS_apply_le

by leonardopedro · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNonnegSemigroup.

BookProof.NonnegSemigroup.norm_semigroupS_apply_le (hT : IsNonnegSelfAdjoint T) (hsv : ∀ w : F, ((0 : F), w) ∈ T → w = 0) {t : ℝ} (ht : 0 ≤ t) (x : F) : ‖semigroupS hT hsv ht x‖ ≤ ‖x‖

Formalization note: Lean 4 identifier BookProof.NonnegSemigroup.norm_semigroupS_apply_le.

Preamble

-- Generated from ChapterNonnegSemigroup.lean — theorem BookProof.NonnegSemigroup.norm_semigroupS_apply_le
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterPositiveSquareRootUnique
import Definitions.Def_ChapterNonnegResolvent
import Definitions.Def_ChapterNonnegUnitaryGroup
import Mathlib
import Definitions.Def_ChapterNonnegSemigroup



open BookProof.ClosureUniqueness
open BookProof.PositiveSquareRoot
open Filter Topology NormedSpace
open scoped InnerProductSpace

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

variable {T : Submodule ℂ (F × F)}

Formal statement

theorem BookProof.NonnegSemigroup.norm_semigroupS_apply_le (hT : IsNonnegSelfAdjoint T)
    (hsv : ∀ w : F, ((0 : F), w) ∈ T → w = 0) {t : ℝ} (ht : 0 ≤ t) (x : F) :
    ‖semigroupS hT hsv ht x‖ ≤ ‖x‖ := by sorry

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