rTensor_injective_iff_subtype — Mathlib

English

If I is IsUnit, then I.FG.

Русский

Если I является IsUnit, то I конечно порождается.

LaTeX

$${\ IsUnit } I \Rightarrow I.FG$$

Lean4

theorem _root_.LinearMap.rTensor_injective_iff_subtype {f : N →ₗ[R] P} (hf : Function.Injective f) (e : P ≃ₗ[R] Q) :
    Function.Injective (f.rTensor M) ↔ Function.Injective ((range <| e.toLinearMap ∘ₗ f).subtype.rTensor M) :=
  by
  simp_rw [← EquivLike.injective_comp <| (LinearEquiv.ofInjective (e.toLinearMap ∘ₗ f) (e.injective.comp hf)).rTensor M,
    ← EquivLike.comp_injective _ (e.rTensor M), ← LinearEquiv.coe_coe, ← coe_comp, LinearEquiv.coe_rTensor, ←
    rTensor_comp]
  rfl

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