ker_sup_ker_le_ker_comp_of_commute

English

Disjoint p (ker f) is equivalent to: for all x ∈ p, f x = 0 implies x = 0.

Русский

Дисjoint p и ker f эквивалентно: для всякого x ∈ p, если f x = 0, то x = 0.

LaTeX

$$Disjoint p (ker f) ↔ ∀ x ∈ p, f x = 0 → x = 0$$

Lean4

theorem ker_sup_ker_le_ker_comp_of_commute {f g : M →ₗ[R] M} (h : Commute f g) : ker f ⊔ ker g ≤ ker (f ∘ₗ g) :=
  by
  refine sup_le_iff.mpr ⟨?_, ker_le_ker_comp g f⟩
  rw [← Module.End.mul_eq_comp, h.eq, Module.End.mul_eq_comp]
  exact ker_le_ker_comp f g

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