prodₗ — Mathlib

English

The product of kernels of two maps is contained in the kernel of their coproduct: (ker f) × (ker g) ≤ ker (f.coprod g).

Русский

Произведение ядер двух отображений содержится в ядре их coprod: (ker f) × (ker g) ⊆ ker (f.coprod g).

LaTeX

$$$$ (\ker f) \times (\ker g) \le \ker(f \coprod g) $$$$

Lean4

/-- `ContinuousLinearMap.prod` as a `LinearEquiv`. -/
@[simps apply]
def prodₗ : ((M →L[R] M₂) × (M →L[R] M₃)) ≃ₗ[S] M →L[R] M₂ × M₃ :=
  { prodEquiv with
    map_add' := fun _f _g => rfl
    map_smul' := fun _c _f => rfl }

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