English
Let f:S → T be a ring homomorphism; then f is injective if and only if for all x,y ∈ R, algebraMap_R_S x = algebraMap_R_S y implies f(algebraMap_R_S x) = f(algebraMap_R_S y).
Русский
Пусть f:S→T — кольцевой гомоморф; тогда f инъективно тогда и только тогда, когда для всех x,y ∈ R, равенство algebraMap_R_S x = algebraMap_R_S y влечет за собой равенство f(algebraMap_R_S x) = f(algebraMap_R_S y).
LaTeX
$$$ \\text{Injective}(f) \\iff \\forall x,y, \\ algebraMap_{R,S}(x) = algebraMap_{R,S}(y) \\Rightarrow f( algebraMap_{R,S}(x) ) = f( algebraMap_{R,S}(y) ). $$$
Lean4
theorem injective_iff_map_algebraMap_eq {T} [CommSemiring T] (f : S →+* T) :
Function.Injective f ↔ ∀ x y, algebraMap R S x = algebraMap R S y ↔ f (algebraMap R S x) = f (algebraMap R S y) :=
by
rw [← IsLocalization.lift_of_comp (M := M) f, IsLocalization.lift_injective_iff]
simp
