span_smul_eq_of_isUnit — Mathlib

English

The coercion map from submodules to sets is ω-Scott continuous with respect to the lattice structure, i.e., it preserves ω-sups along increasing chains.

Русский

Сопоставление подмодулей и множеств сохраняет ω-поиск по возрастающим цепям.

LaTeX

$$$\Omega\text{-complete partial order}.ω\text{ScottContinuous}(\uparrow) $ для $\text{Submodule} \to \text{Set}$$$

Lean4

/-- See `Submodule.span_smul_eq` (in `RingTheory.Ideal.Operations`) for
`span R (r • s) = r • span R s` that holds for arbitrary `r` in a `CommSemiring`. -/
theorem span_smul_eq_of_isUnit (s : Set M) (r : R) (hr : IsUnit r) : span R (r • s) = span R s :=
  by
  apply le_antisymm
  · apply span_smul_le
  · convert span_smul_le (r • s) ((hr.unit⁻¹ :) : R)
    simp [smul_smul]

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