smul_balancedCore_subset — Mathlib

English

If ||a|| ≤ 1, then a · balancedCore 𝕜 s ⊆ balancedCore 𝕜 s.

Русский

Если ||a|| ≤ 1, то a · balancedCore 𝕜 s ⊆ balancedCore 𝕜 s.

LaTeX

$$$a \\cdot balancedCore 𝕜 s ⊆ balancedCore 𝕜 s \\quad\\text{when } \\|a\\| \\le 1$$$

Lean4

theorem smul_balancedCore_subset (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) : a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s :=
  by
  rintro x ⟨y, hy, rfl⟩
  rw [mem_balancedCore_iff] at hy
  rcases hy with ⟨t, ht1, ht2, hy⟩
  exact ⟨t, ⟨ht1, ht2⟩, ht1 a ha (smul_mem_smul_set hy)⟩

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