Theorems · Theorem · group theory

AddSubmonoid.LocalizationMap.map_injective_of_injective

∀ {M : Type u_1} [inst : AddCommMonoid M] {S : AddSubmonoid M} {N : Type u_2} [inst_1 : AddCommMonoid N] {P : Type u_3}
  [inst_2 : AddCommMonoid P] (f : S.LocalizationMap N) {g : M →+ P} {Q : Type u_4} [inst_3 : AddCommMonoid Q],
  Function.Injective ⇑g → ∀ (k : (AddSubmonoid.map g S).LocalizationMap Q), Function.Injective ⇑(f.map ⋯ k)

Given an injective AddCommMonoid homomorphism g : M →+ P, and a submonoid S ⊆ M, the induced monoid homomorphism from the localization of M at S to the localization of P at g S, is injective.

Defined in

Mathlib.GroupTheory.MonoidLocalization.Maps

Cited by

0 results in Mathlib

Foundations

Depth 52 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidAddCommMonoidAddCommMonoidAddCommMonoid