restrict_scalars — Mathlib

English

Absorbs is preserved under restriction of scalars along a scalar tower if a cobounded map tends from M to N.

Русский

Поглощение сохраняется при ограничении галуальностей вдоль тензорной башни скаляров при подходящей предельности.

LaTeX

$$$Absorbs\\ M\\ s\\ t \\land Tendsto (\\cdot \\cdot 1) (cobounded M) (cobounded N) \\Rightarrow Absorbs M s t$$$

Lean4

protected theorem restrict_scalars {M N α : Type*} [Monoid N] [SMul M N] [SMul M α] [MulAction N α]
    [IsScalarTower M N α] [Bornology M] [Bornology N] {s t : Set α} (h : Absorbs N s t)
    (hbdd : Tendsto (· • 1 : M → N) (cobounded M) (cobounded N)) : Absorbs M s t :=
  (hbdd.eventually h).mono <| fun x hx ↦ by rwa [smul_one_smul N x s] at hx

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