kerLift_injective — Mathlib

English

The map kerLift from M to P is injective for any fixed f; i.e., kerLift is an embedding on the quotient.

Русский

Отображение kerLift инъективно на каждом фиксированном f; то есть ядро-лифт — вложение в искомый множ.

LaTeX

$$$\text{kerLift injective}$$$

Lean4

/-- A monoid homomorphism `f` induces an injective homomorphism on the quotient by `f`'s kernel. -/
@[to_additive /-- An `AddMonoid` homomorphism `f` induces an injective homomorphism on the quotient
    by `f`'s kernel. -/
    ]
theorem kerLift_injective (f : M →* P) : Injective (kerLift f) := fun x y =>
  Quotient.inductionOn₂' x y fun _ _ => (ker f).eq.2

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