isCompl_dualAnnihilator — Mathlib

English

If s < t are subspaces of a finite-dimensional vector space over K, then finrank(s) < finrank(t).

Русский

Если s и t — подпространства в конечномерном векторном пространстве над K и s вложено в t, то finrank(s) < finrank(t).

LaTeX

$$$ s < t \\implies \\operatorname{finrank}_K(s) < \\operatorname{finrank}_K(t) $$$

Lean4

/-- For vector spaces, dual annihilators carry direct sum decompositions
to direct sum decompositions. -/
theorem isCompl_dualAnnihilator {W W' : Subspace K V₁} (h : IsCompl W W') :
    IsCompl W.dualAnnihilator W'.dualAnnihilator :=
  by
  rw [isCompl_iff, disjoint_iff, codisjoint_iff] at h ⊢
  rw [← dualAnnihilator_inf_eq, ← dualAnnihilator_sup_eq, h.1, h.2, dualAnnihilator_top, dualAnnihilator_bot]
  exact ⟨rfl, rfl⟩

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