toSubmoduleEquiv — Mathlib

English

The image of a subalgebra S of A under an algebra homomorphism f: A →ₐ[R] B is a subalgebra of B; more precisely, the carrier of map f applied to S is the set { f(a) : a ∈ S } which forms a subalgebra of B.

Русский

Образ подалгебры S ⊆ A под однородным гомоморфизмом f: A →ₐ[R] B образует подалгебру в B; множество { f(a) : a ∈ S } образует подалгебру в B.

LaTeX

$$$\operatorname{carrier}(\operatorname{map}(f,S)) = \{ f(a) : a \in S \}$$$

Lean4

/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,
we define it as a `LinearEquiv` to avoid type equalities. -/
def toSubmoduleEquiv (S : Subalgebra R A) : toSubmodule S ≃ₗ[R] S :=
  LinearEquiv.ofEq _ _ rfl

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