finrank_vectorSpan_add_one — Mathlib

English

The vector span of a finite subset has a well-defined finite dimension.

Русский

ВекторSpan конечного подмножества имеет конечную размерность.

LaTeX

$$$\operatorname{finrank}_k\bigl(\operatorname{vectorSpan}_k(\operatorname{SetRange}(p))\bigr) = n$ для некоторого n$$

Lean4

/-- The `vectorSpan` of a finite affinely independent family has dimension one less than its
cardinality. -/
theorem finrank_vectorSpan_add_one [Fintype ι] [Nonempty ι] {p : ι → P} (hi : AffineIndependent k p) :
    finrank k (vectorSpan k (Set.range p)) + 1 = Fintype.card ι := by
  rw [hi.finrank_vectorSpan (tsub_add_cancel_of_le _).symm, tsub_add_cancel_of_le] <;> exact Fintype.card_pos

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