coeFn_sum — Mathlib

English

Let N be a submodule of M and G an endomorphism of M with G^k = 0 and a compatibility condition G∘N.subtype = N.subtype∘g for a g endomorphism of N. Then g^k = 0.

Русский

Пусть N — подпространство M и G — линейное отображение M, при условии G^k = 0 и совместимости G∘N.subtype = N.subtype∘g, então g^k = 0.

LaTeX

$$$G^k=0\\Rightarrow g^k=0 \\text{ given } G\\circ N=.\\text{subtype} = N.\\text{subtype} \\circ g.$$$

Lean4

@[simp, norm_cast]
theorem coeFn_sum {ι : Type*} (t : Finset ι) (f : ι → M →ₛₗ[σ₁₂] M₂) : ⇑(∑ i ∈ t, f i) = ∑ i ∈ t, (f i : M → M₂) :=
  _root_.map_sum
    (show AddMonoidHom (M →ₛₗ[σ₁₂] M₂) (M → M₂) from
      { toFun := DFunLike.coe, map_zero' := rfl
        map_add' := fun _ _ => rfl })
    _ _

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