Theorems · Theorem · group theory

Submonoid.LocalizationMap.map_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} {T : Submonoid P} (hy : ∀ (y : ↥S), g ↑y ∈ T)
  {Q : Type u_4} [inst_3 : CommMonoid Q] {k : T.LocalizationMap Q} (z : N) (u : Q),
  (f.map hy k) z = u ↔ k (g (f.sec z).1) = k (g ↑(f.sec z).2) * u

Given Localization maps f : M →* N, k : P →* Q for Submonoids S, T respectively, if a CommMonoid homomorphism g : M →* P induces a f.map hy k : N →* Q, then for all z : N, u : Q, we have f.map hy k z = u ↔ k (g x) = k (g y) * u where x : M, y ∈ S are such that z * f y = f x.

Defined in

Mathlib.GroupTheory.MonoidLocalization.Maps

Cited by

0 results in Mathlib

Foundations

Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidCommMonoidCommMonoidCommMonoid