English
The equality-based isometry on a subspace with hpq = rfl is the identity isometry on that subspace.
Русский
Изометрия, заданная равенством в случае rfl, является тождественной изометрией на подпространстве.
LaTeX
$$$ofEq p p rfl = \mathrm{LinearIsometryEquiv.refl} R' p$$$
Lean4
/-- Reinterpret a `LinearIsometry` as a `LinearIsometryEquiv` to the range. -/
@[simps! apply_coe]
noncomputable def equivRange {R S : Type*} [Semiring R] [Ring S] [Module S E] [Module R F] {σ₁₂ : R →+* S}
{σ₂₁ : S →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : F →ₛₗᵢ[σ₁₂] E) :
F ≃ₛₗᵢ[σ₁₂] (LinearMap.range f.toLinearMap) :=
{ f with toLinearEquiv := LinearEquiv.ofInjective f.toLinearMap f.injective }