surjective_of_injective — Mathlib

English

If f: V → V₂ is surjective and V is finite-dimensional, then V₂ is finite-dimensional.

Русский

Пусть f: V→V₂ сюръективно и V конечномерно; тогда V₂ тоже конечномерно.

LaTeX

$$$[\operatorname{FiniteDimensional}_K V] \Rightarrow (\operatorname{range} f = \top) \Rightarrow \operatorname{FiniteDimensional}_K V_2$$$

Lean4

/-- On a finite-dimensional space, an injective linear map is surjective. -/
theorem surjective_of_injective [FiniteDimensional K V] {f : V →ₗ[K] V} (hinj : Injective f) : Surjective f :=
  by
  have h := rank_range_of_injective _ hinj
  rw [← finrank_eq_rank, ← finrank_eq_rank, Nat.cast_inj] at h
  exact range_eq_top.1 (eq_top_of_finrank_eq h)

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