toNonUnitalSubring_injective — Mathlib

English

The map that takes a nonunital star-subalgebra S to its underlying nonunital subring is injective; equivalently, if S.toNonUnitalSubring = T.toNonUnitalSubring, then S = T.

Русский

Картографирование, отправляющее неполную звездную подпалгебру S в соответствующее неполное кольцо S.toNonUnitalSubring, инъективно; то есть если S.toNonUnitalSubring = T.toNonUnitalSubring, то S = T.

LaTeX

$$The function S ↦ S.toNonUnitalSubring is injective: S.toNonUnitalSubring = T.toNonUnitalSubring ⇒ S = T.$$

Lean4

theorem toNonUnitalSubring_injective {R : Type u} {A : Type v} [CommRing R] [NonUnitalRing A] [Module R A] [Star A] :
    Function.Injective (toNonUnitalSubring : NonUnitalStarSubalgebra R A → NonUnitalSubring A) := fun S T h =>
  ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]

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