exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card

English

In the same setting, a submodule S with maximal finrank equals the whole space.

Русский

В той же настройке подпространство S с максимальной размерностью совпадает с пространством целиком.

LaTeX

$$$ S = \top $ вслед за $ finrank K S = finrank K V $$$

Lean4

/-- A slight strengthening of `exists_nontrivial_relation_sum_zero_of_rank_succ_lt_card`
available when working over an ordered field:
we can ensure a positive coefficient, not just a nonzero coefficient.
-/
theorem exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card [FiniteDimensional L W] {t : Finset W}
    (h : finrank L W + 1 < t.card) : ∃ f : W → L, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x :=
  by
  obtain ⟨f, sum, total, nonzero⟩ := Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card h
  exact ⟨f, sum, total, exists_pos_of_sum_zero_of_exists_nonzero f total nonzero⟩

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