linearMap_compatibleSMul — Mathlib

English

For a localization map f: S → Q with hf = algebraMap, f is bijective.

Русский

Для локализационной карты f: S → Q с hf = алгебраMap, карта биекция.

LaTeX

$$$\\text{Bijective}(f)$, given $f\\circ \\text{algebraMap}_R^S = \\text{algebraMap}_R^Q$$$

Lean4

theorem linearMap_compatibleSMul (N₁ N₂) [AddCommMonoid N₁] [AddCommMonoid N₂] [Module R N₁] [Module S N₁] [Module R N₂]
    [Module S N₂] [IsScalarTower R S N₁] [IsScalarTower R S N₂] : LinearMap.CompatibleSMul N₁ N₂ S R where
  map_smul f s
    s' := by
    obtain ⟨r, m, rfl⟩ := mk'_surjective M s
    rw [← (map_units S m).smul_left_cancel]
    simp_rw [algebraMap_smul, ← map_smul, ← smul_assoc, smul_mk'_self, algebraMap_smul, map_smul]

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