map_injective — Mathlib

English

As in 132808, the submodule equals span of the representative.

Русский

Как и ранее, подпространство равно span представителя.

LaTeX

$$$ v.\mathrm{submodule} = \operatorname{span}_K\{ v.rep \} $$$

Lean4

/-- Mapping with respect to a semilinear map over an isomorphism of fields yields
an injective map on projective spaces. -/
theorem map_injective {σ : K →+* L} {τ : L →+* K} [RingHomInvPair σ τ] (f : V →ₛₗ[σ] W) (hf : Function.Injective f) :
    Function.Injective (map f hf) := fun u v h ↦ by
  induction u using ind with
  | h u hu =>
    induction v using ind with
    | h v hv =>
      simp only [map_mk, mk_eq_mk_iff'] at h ⊢
      rcases h with ⟨a, ha⟩
      refine ⟨τ a, hf ?_⟩
      rwa [f.map_smulₛₗ, RingHomInvPair.comp_apply_eq₂]

Похожие утверждения