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[Merged by Bors] - feat(analysis/normed_space/linear_isometry): linear_equiv.of_eq as a linear_isometry_equiv #15471

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[Merged by Bors] - feat(analysis/normed_space/linear_isometry): linear_equiv.of_eq as a linear_isometry_equiv #15471

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`linear_equiv.of_eq` as a `linear_isometry_equiv`
ADedecker Jul 17, 2022
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WIP
ADedecker Jul 18, 2022
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ADedecker Jul 20, 2022
0539171
Fix
ADedecker Jul 20, 2022
cba6d45
No @[simps]
ADedecker Jul 20, 2022
80871bd
Fix
ADedecker Jul 20, 2022
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14 changes: 14 additions & 0 deletions src/analysis/normed_space/linear_isometry.lean
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Expand Up @@ -586,6 +586,20 @@ rfl
((prod_assoc R E E₂ E₃).symm : E × E₂ × E₃ → (E × E₂) × E₃) = (equiv.prod_assoc E E₂ E₃).symm :=
rfl

variables {R E E₂ E₃} {R' : Type*} [ring R'] [module R' E] (p q : submodule R' E)

/-- `linear_equiv.of_eq` as a `linear_isometry_equiv`. -/
def of_eq (hpq : p = q) :
p ≃ₗᵢ[R'] q :=
{ norm_map' := λ x, rfl,
..linear_equiv.of_eq p q hpq }

variables {p q}

@[simp] lemma coe_of_eq_apply (h : p = q) (x : p) : (of_eq p q h x : E) = x := rfl
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Could you please setup simps to work with linear isometries, if that is not already done? Then these lemmas can be auto-generated.

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I don't think simps is easy to configure to generate the right name here (with coe as a prefix)

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Yes, if we want to autogenerate these we can't do anything about the names. I'm struggling a bit to understand how to setup these properly, but I'll ask people in Providence tomorrow

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I did the simps setup and removed the coe_of_eq_apply, here and for linear_equiv, since simps generate it as of_eq_apply, which is more homogeneous with the rest of the library. Does that sound fine ?


@[simp] lemma of_eq_symm (h : p = q) : (of_eq p q h).symm = of_eq q p h.symm := rfl
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end linear_isometry_equiv

/-- Two linear isometries are equal if they are equal on basis vectors. -/
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