span_eq_top_iff — Mathlib

English

The pair (map f, comap f) forms a Galois connection between subspaces via f.

Русский

Пара (map f, comap f) образует связь Гало между подпроизвольствами через f.

LaTeX

$$$\\\\mathrm{GaloisConnection}(\\\\mathrm{map}\\\\ f,\\\\mathrm{comap}\\\\ f)$$$

Lean4

theorem span_eq_top_iff {s : Set P₁} (e : P₁ ≃ᵃ[k] P₂) : affineSpan k s = ⊤ ↔ affineSpan k (e '' s) = ⊤ :=
  by
  refine ⟨(e : P₁ →ᵃ[k] P₂).span_eq_top_of_surjective e.surjective, ?_⟩
  intro h
  have : s = e.symm '' (e '' s) := by rw [← image_comp]; simp
  rw [this]
  exact (e.symm : P₂ →ᵃ[k] P₁).span_eq_top_of_surjective e.symm.surjective h

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