range_prod_eq — Mathlib

English

Let L be a linear map from the product ∏i φ_i to M. Then summing the compositions with each Pi.single i recovers L, i.e., ∑ i L ∘ (single i) (v_i) = L(v).

Русский

Пусть L: (∏i φ_i) → M линейно. Тогда сумма по i от L ∘ (single i) applied к v_i равна L(v).

LaTeX

$$$$ \sum_i L.comp(\text{single } R \; \varphi \; i) (v_i) = L(v) $$$$

Lean4

theorem range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃} (h : ker f ⊔ ker g = ⊤) :
    range (f.prod g) = (range f).prod (range g) :=
  LinearMap.range_prod_eq h

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