faithfulSMul — Mathlib

English

A cancellative monoid acts faithfully on itself by left multiplication.

Русский

Относительно левостороннего умножения верно действие подписанного моноида на сам монойд.

LaTeX

$$$\\text{FaithfulSMul } \\alpha \\alpha:$ the map $a \\mapsto (x \\mapsto a\\cdot x)$ is injective.$$

Lean4

/-- `Monoid.toMulAction` is faithful on cancellative monoids. -/
@[to_additive /-- `AddMonoid.toAddAction` is faithful on additive cancellative monoids. -/
    ]
instance faithfulSMul [RightCancelMonoid α] : FaithfulSMul α α :=
  ⟨fun h ↦ mul_right_cancel (h 1)⟩

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