linearEquivOfRightInverse — Mathlib

English

If f is a linear map with right inverse, then there exists a linear equivalence M ≃ N.

Русский

Если f имеет правое обратное, существует линейное эквив между M и N.

LaTeX

$$RightInverse f g → ∃ e: M ≃ N$$

Lean4

/-- If `f : M →ₗ[R] N` and `g : N →ₗ[R] M` where `M` and `N` are invertible `R`-modules, and `f` is
a right inverse of `g`, then in fact `f` is also the left inverse of `g`, and we promote this to
an `R`-module isomorphism. -/
def linearEquivOfRightInverse (hfg : Function.RightInverse f g) : M ≃ₗ[R] N :=
  .ofLinear f g (LinearMap.ext <| leftInverse_of_rightInverse hfg) (LinearMap.ext hfg)

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