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[Merged by Bors] - chore(Analysis/Complex/Circle): nnnorm and enorm lemmas for circle #43053

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[Merged by Bors] - chore(Analysis/Complex/Circle): nnnorm and enorm lemmas for circle #43053

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22 changes: 15 additions & 7 deletions Mathlib/Analysis/Complex/Circle.lean
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Expand Up @@ -71,7 +71,9 @@ lemma coe_injective : Injective ((↑) : Circle → ℂ) := fun _ _ ↦ ext
-- Not simp because `SetLike.coe_eq_coe` already proves it
lemma coe_inj : (x : ℂ) = y ↔ x = y := coe_injective.eq_iff

lemma norm_coe (z : Circle) : ‖(z : ℂ)‖ = 1 := mem_sphere_zero_iff_norm.1 z.2
@[simp] lemma norm_coe (z : Circle) : ‖(z : ℂ)‖ = 1 := mem_sphere_zero_iff_norm.1 z.2
@[simp] lemma nnnorm_coe (z : Circle) : ‖(z : ℂ)‖₊ = 1 := NNReal.coe_injective z.norm_coe
@[simp] lemma enorm_coe (z : Circle) : ‖(z : ℂ)‖ₑ = 1 := by simp [enorm_eq_nnnorm]

set_option backward.isDefEq.respectTransparency false in
@[simp] lemma normSq_coe (z : Circle) : normSq z = 1 := by simp [normSq_eq_norm_sq]
Expand Down Expand Up @@ -198,12 +200,18 @@ instance instContinuousSMul [TopologicalSpace α] [MulAction ℂ α] [Continuous
ContinuousSMul Circle α :=
inferInstanceAs <| ContinuousSMul (Submonoid.unitSphere _) α

set_option backward.isDefEq.respectTransparency false in
@[simp]
protected lemma norm_smul {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℂ E]
(u : Circle) (v : E) :
‖u • v‖ = ‖v‖ := by
rw [smul_def, norm_smul, norm_eq_of_mem_sphere, one_mul]
section Norm

variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℂ E] (u : Circle) (v : E)

@[simp] protected lemma norm_smul : ‖u • v‖ = ‖v‖ := by simp [smul_def, norm_smul]
@[simp] protected lemma nnnorm_smul : ‖u • v‖₊ = ‖v‖₊ := NNReal.coe_injective (u.norm_smul v)
@[simp] protected lemma enorm_smul : ‖u • v‖ₑ = ‖v‖ₑ := by simp [enorm_eq_nnnorm]

instance : IsIsometricSMul Circle E :=
⟨fun _ ↦ Isometry.of_dist_eq fun _ _ ↦ by simp [dist_eq_norm, ← smul_sub]⟩

end Norm

end Circle

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