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[Merged by Bors] - feat(topology/metric_space/basic): some lemmas about dist #13343

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[Merged by Bors] - feat(topology/metric_space/basic): some lemmas about dist #13343

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1a4a4b4
some lemmas
fpvandoorn Apr 8, 2022
11356e9
Merge remote-tracking branch 'origin/master' into fpvandoorn/dist
fpvandoorn Apr 11, 2022
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6 changes: 6 additions & 0 deletions src/analysis/normed/group/basic.lean
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Expand Up @@ -152,6 +152,12 @@ by simp only [sub_eq_add_neg, dist_add_left, dist_neg_neg]
@[simp] lemma dist_sub_right (g₁ g₂ h : E) : dist (g₁ - h) (g₂ - h) = dist g₁ g₂ :=
by simpa only [sub_eq_add_neg] using dist_add_right _ _ _

@[simp] theorem dist_self_add_right (g h : E) : dist g (g + h) = ∥h∥ :=
by rw [← dist_zero_left, ← dist_add_left g 0 h, add_zero]

@[simp] theorem dist_self_add_left (g h : E) : dist (g + h) g = ∥h∥ :=
by rw [dist_comm, dist_self_add_right]

/-- **Triangle inequality** for the norm. -/
lemma norm_add_le (g h : E) : ∥g + h∥ ≤ ∥g∥ + ∥h∥ :=
by simpa [dist_eq_norm] using dist_triangle g 0 (-h)
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17 changes: 13 additions & 4 deletions src/topology/metric_space/basic.lean
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Expand Up @@ -1404,8 +1404,9 @@ end
end nnreal

section prod
variables [pseudo_metric_space β]

noncomputable instance prod.pseudo_metric_space_max [pseudo_metric_space β] :
noncomputable instance prod.pseudo_metric_space_max :
pseudo_metric_space (α × β) :=
{ dist := λ x y, max (dist x.1 y.1) (dist x.2 y.2),
dist_self := λ x, by simp,
Expand All @@ -1427,14 +1428,22 @@ noncomputable instance prod.pseudo_metric_space_max [pseudo_metric_space β] :
end,
to_uniform_space := prod.uniform_space }

lemma prod.dist_eq [pseudo_metric_space β] {x y : α × β} :
lemma prod.dist_eq {x y : α × β} :
dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl

theorem ball_prod_same [pseudo_metric_space β] (x : α) (y : β) (r : ℝ) :
@[simp]
lemma dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ :=
by simp [prod.dist_eq, dist_nonneg]

@[simp]
lemma dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ :=
by simp [prod.dist_eq, dist_nonneg]

theorem ball_prod_same (x : α) (y : β) (r : ℝ) :
ball x r ×ˢ ball y r = ball (x, y) r :=
ext $ λ z, by simp [prod.dist_eq]

theorem closed_ball_prod_same [pseudo_metric_space β] (x : α) (y : β) (r : ℝ) :
theorem closed_ball_prod_same (x : α) (y : β) (r : ℝ) :
closed_ball x r ×ˢ closed_ball y r = closed_ball (x, y) r :=
ext $ λ z, by simp [prod.dist_eq]

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