surjective_compr₂ₛₗ_of_exists_rightInverse

English

If f is bijective and g is a linear equivalence, then f.compr₂ g is bijective.

Русский

Если f биективна и g линейная эквивалентность, то f.compr₂ g биективна.

LaTeX

$$$\text{Bijective}(f) \Rightarrow \text{Bijective}(f.compr₂ g.toLinearMap).$$$

Lean4

/-- A version of `Function.Surjective.comp` for composition of a bilinear map with a linear map. -/
theorem surjective_compr₂ₛₗ_of_exists_rightInverse [RingHomInvPair σ₃₄ σ₄₃] (f : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P)
    (g : P →ₛₗ[σ₃₄] Q) (hf : Surjective f) (hg : ∃ g' : Q →ₛₗ[σ₄₃] P, g.comp g' = LinearMap.id) :
    Surjective (f.compr₂ₛₗ g) :=
  (surjective_comp_left_of_exists_rightInverse hg).comp hf

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