Theorems · Theorem · group theory

Submonoid.LocalizationMap.lift_spec

∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {P : Type u_3}
  [inst_2 : CommMonoid P] (f : S.LocalizationMap N) {g : M →* P} (hg : ∀ (y : ↥S), IsUnit (g ↑y)) (z : N) (v : P),
  (f.lift hg) z = v ↔ g (f.sec z).1 = g ↑(f.sec z).2 * v

Given a Localization map f : M →* N for a Submonoid S ⊆ M, if a CommMonoid map g : M →* P induces a map f.lift hg : N →* P then for all z : N, v : P, we have f.lift hg z = v ↔ g x = g y * v, where x : M, y ∈ S are such that z * f y = f x.

Defined in

Mathlib.GroupTheory.MonoidLocalization.Maps

Cited by

3 results in Mathlib

Foundations

Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidCommMonoidCommMonoid