File: Mathlib/Analysis/Normed/Group/Pointwise.lean

full_name: Bornology.IsBounded.mul
code: @[to_additive]
theorem Bornology.IsBounded.mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) 
goal: Bornology.IsBounded (s * t)

full_name: Bornology.IsBounded.add
code: @[to_additive]
theorem Bornology.IsBounded.mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) 
goal: Bornology.IsBounded (s + t)

full_name: Bornology.IsBounded.of_add
code: @[to_additive]
theorem Bornology.IsBounded.of_mul (hst : IsBounded (s * t)) : IsBounded s ∨ IsBounded t 
goal: Bornology.IsBounded s ∨ Bornology.IsBounded t

full_name: Bornology.IsBounded.of_mul
code: @[to_additive]
theorem Bornology.IsBounded.of_mul (hst : IsBounded (s * t)) : IsBounded s ∨ IsBounded t 
goal: Bornology.IsBounded s ∨ Bornology.IsBounded t

full_name: Bornology.IsBounded.neg
code: @[to_additive]
theorem Bornology.IsBounded.inv : IsBounded s → IsBounded s⁻¹ 
goal: Bornology.IsBounded s → Bornology.IsBounded (-s)

full_name: Bornology.IsBounded.inv
code: @[to_additive]
theorem Bornology.IsBounded.inv : IsBounded s → IsBounded s⁻¹ 
goal: Bornology.IsBounded s → Bornology.IsBounded s⁻¹

full_name: Bornology.IsBounded.div
code: @[to_additive]
theorem Bornology.IsBounded.div (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s / t) 
goal: Bornology.IsBounded (s / t)

full_name: Bornology.IsBounded.sub
code: @[to_additive]
theorem Bornology.IsBounded.div (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s / t) 
goal: Bornology.IsBounded (s - t)

full_name: infEdist_neg_neg
code: @[to_additive (attr := simp)]
theorem infEdist_inv_inv (x : E) (s : Set E) : infEdist x⁻¹ s⁻¹ = infEdist x s 
goal: EMetric.infEdist (-x) (-s) = EMetric.infEdist x s

full_name: infEdist_inv_inv
code: @[to_additive (attr := simp)]
theorem infEdist_inv_inv (x : E) (s : Set E) : infEdist x⁻¹ s⁻¹ = infEdist x s 
goal: EMetric.infEdist x⁻¹ s⁻¹ = EMetric.infEdist x s

full_name: infEdist_neg
code: @[to_additive]
theorem infEdist_inv (x : E) (s : Set E) : infEdist x⁻¹ s = infEdist x s⁻¹ 
goal: EMetric.infEdist (-x) s = EMetric.infEdist x (-s)

full_name: infEdist_inv
code: @[to_additive]
theorem infEdist_inv (x : E) (s : Set E) : infEdist x⁻¹ s = infEdist x s⁻¹ 
goal: EMetric.infEdist x⁻¹ s = EMetric.infEdist x s⁻¹

full_name: ediam_mul_le
code: @[to_additive]
theorem ediam_mul_le (x y : Set E) : EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y 
goal: EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y

full_name: ediam_add_le
code: @[to_additive]
theorem ediam_mul_le (x y : Set E) : EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y 
goal: EMetric.diam (x + y) ≤ EMetric.diam x + EMetric.diam y

full_name: neg_thickening
code: @[to_additive (attr := simp)]
theorem inv_thickening : (thickening δ s)⁻¹ = thickening δ s⁻¹ 
goal: -Metric.thickening δ s = Metric.thickening δ (-s)

full_name: inv_thickening
code: @[to_additive (attr := simp)]
theorem inv_thickening : (thickening δ s)⁻¹ = thickening δ s⁻¹ 
goal: (Metric.thickening δ s)⁻¹ = Metric.thickening δ s⁻¹

full_name: neg_cthickening
code: @[to_additive (attr := simp)]
theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ 
goal: -Metric.cthickening δ s = Metric.cthickening δ (-s)

full_name: inv_cthickening
code: @[to_additive (attr := simp)]
theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ 
goal: (Metric.cthickening δ s)⁻¹ = Metric.cthickening δ s⁻¹

full_name: inv_ball
code: @[to_additive (attr := simp)]
theorem inv_ball : (ball x δ)⁻¹ = ball x⁻¹ δ 
goal: (Metric.ball x δ)⁻¹ = Metric.ball x⁻¹ δ

full_name: neg_ball
code: @[to_additive (attr := simp)]
theorem inv_ball : (ball x δ)⁻¹ = ball x⁻¹ δ 
goal: -Metric.ball x δ = Metric.ball (-x) δ

full_name: neg_closedBall
code: @[to_additive (attr := simp)]
theorem inv_closedBall : (closedBall x δ)⁻¹ = closedBall x⁻¹ δ 
goal: -Metric.closedBall x δ = Metric.closedBall (-x) δ

full_name: inv_closedBall
code: @[to_additive (attr := simp)]
theorem inv_closedBall : (closedBall x δ)⁻¹ = closedBall x⁻¹ δ 
goal: (Metric.closedBall x δ)⁻¹ = Metric.closedBall x⁻¹ δ

full_name: singleton_add_ball
code: @[to_additive]
theorem singleton_mul_ball : {x} * ball y δ = ball (x * y) δ 
goal: {x} + Metric.ball y δ = Metric.ball (x + y) δ

full_name: singleton_mul_ball
code: @[to_additive]
theorem singleton_mul_ball : {x} * ball y δ = ball (x * y) δ 
goal: {x} * Metric.ball y δ = Metric.ball (x * y) δ

full_name: singleton_sub_ball
code: @[to_additive]
theorem singleton_div_ball : {x} / ball y δ = ball (x / y) δ 
goal: {x} - Metric.ball y δ = Metric.ball (x - y) δ

full_name: singleton_div_ball
code: @[to_additive]
theorem singleton_div_ball : {x} / ball y δ = ball (x / y) δ 
goal: {x} / Metric.ball y δ = Metric.ball (x / y) δ

full_name: ball_add_singleton
code: @[to_additive]
theorem ball_mul_singleton : ball x δ * {y} = ball (x * y) δ 
goal: Metric.ball x δ + {y} = Metric.ball (x + y) δ

full_name: ball_mul_singleton
code: @[to_additive]
theorem ball_mul_singleton : ball x δ * {y} = ball (x * y) δ 
goal: Metric.ball x δ * {y} = Metric.ball (x * y) δ

full_name: ball_div_singleton
code: @[to_additive]
theorem ball_div_singleton : ball x δ / {y} = ball (x / y) δ 
goal: Metric.ball x δ / {y} = Metric.ball (x / y) δ

full_name: ball_sub_singleton
code: @[to_additive]
theorem ball_div_singleton : ball x δ / {y} = ball (x / y) δ 
goal: Metric.ball x δ - {y} = Metric.ball (x - y) δ

full_name: singleton_add_ball_zero
code: @[to_additive]
theorem singleton_mul_ball_one : {x} * ball 1 δ = ball x δ 
goal: {x} + Metric.ball 0 δ = Metric.ball x δ

full_name: singleton_mul_ball_one
code: @[to_additive]
theorem singleton_mul_ball_one : {x} * ball 1 δ = ball x δ 
goal: {x} * Metric.ball 1 δ = Metric.ball x δ

full_name: singleton_div_ball_one
code: @[to_additive]
theorem singleton_div_ball_one : {x} / ball 1 δ = ball x δ 
goal: {x} / Metric.ball 1 δ = Metric.ball x δ

full_name: singleton_sub_ball_zero
code: @[to_additive]
theorem singleton_div_ball_one : {x} / ball 1 δ = ball x δ 
goal: {x} - Metric.ball 0 δ = Metric.ball x δ

full_name: ball_one_mul_singleton
code: @[to_additive]
theorem ball_one_mul_singleton : ball 1 δ * {x} = ball x δ 
goal: Metric.ball 1 δ * {x} = Metric.ball x δ

full_name: ball_zero_add_singleton
code: @[to_additive]
theorem ball_one_mul_singleton : ball 1 δ * {x} = ball x δ 
goal: Metric.ball 0 δ + {x} = Metric.ball x δ

full_name: ball_zero_sub_singleton
code: @[to_additive]
theorem ball_one_div_singleton : ball 1 δ / {x} = ball x⁻¹ δ 
goal: Metric.ball 0 δ - {x} = Metric.ball (-x) δ

full_name: ball_one_div_singleton
code: @[to_additive]
theorem ball_one_div_singleton : ball 1 δ / {x} = ball x⁻¹ δ 
goal: Metric.ball 1 δ / {x} = Metric.ball x⁻¹ δ

full_name: vadd_ball_zero
code: @[to_additive]
theorem smul_ball_one : x • ball (1 : E) δ = ball x δ 
goal: x +ᵥ Metric.ball 0 δ = Metric.ball x δ

full_name: smul_ball_one
code: @[to_additive]
theorem smul_ball_one : x • ball (1 : E) δ = ball x δ 
goal: x • Metric.ball 1 δ = Metric.ball x δ

full_name: singleton_mul_closedBall
code: @[to_additive (attr := simp 1100)]
theorem singleton_mul_closedBall : {x} * closedBall y δ = closedBall (x * y) δ 
goal: {x} * Metric.closedBall y δ = Metric.closedBall (x * y) δ

full_name: singleton_add_closedBall
code: @[to_additive (attr := simp 1100)]
theorem singleton_mul_closedBall : {x} * closedBall y δ = closedBall (x * y) δ 
goal: {x} + Metric.closedBall y δ = Metric.closedBall (x + y) δ

full_name: singleton_sub_closedBall
code: @[to_additive (attr := simp 1100)]
theorem singleton_div_closedBall : {x} / closedBall y δ = closedBall (x / y) δ 
goal: {x} - Metric.closedBall y δ = Metric.closedBall (x - y) δ

full_name: singleton_div_closedBall
code: @[to_additive (attr := simp 1100)]
theorem singleton_div_closedBall : {x} / closedBall y δ = closedBall (x / y) δ 
goal: {x} / Metric.closedBall y δ = Metric.closedBall (x / y) δ

full_name: closedBall_mul_singleton
code: @[to_additive (attr := simp 1100)]
theorem closedBall_mul_singleton : closedBall x δ * {y} = closedBall (x * y) δ 
goal: Metric.closedBall x δ * {y} = Metric.closedBall (x * y) δ

full_name: closedBall_add_singleton
code: @[to_additive (attr := simp 1100)]
theorem closedBall_mul_singleton : closedBall x δ * {y} = closedBall (x * y) δ 
goal: Metric.closedBall x δ + {y} = Metric.closedBall (x + y) δ

full_name: closedBall_div_singleton
code: @[to_additive (attr := simp 1100)]
theorem closedBall_div_singleton : closedBall x δ / {y} = closedBall (x / y) δ 
goal: Metric.closedBall x δ / {y} = Metric.closedBall (x / y) δ

full_name: closedBall_sub_singleton
code: @[to_additive (attr := simp 1100)]
theorem closedBall_div_singleton : closedBall x δ / {y} = closedBall (x / y) δ 
goal: Metric.closedBall x δ - {y} = Metric.closedBall (x - y) δ

full_name: singleton_add_closedBall_zero
code: @[to_additive]
theorem singleton_mul_closedBall_one : {x} * closedBall 1 δ = closedBall x δ 
goal: {x} + Metric.closedBall 0 δ = Metric.closedBall x δ

full_name: singleton_mul_closedBall_one
code: @[to_additive]
theorem singleton_mul_closedBall_one : {x} * closedBall 1 δ = closedBall x δ 
goal: {x} * Metric.closedBall 1 δ = Metric.closedBall x δ

full_name: singleton_sub_closedBall_zero
code: @[to_additive]
theorem singleton_div_closedBall_one : {x} / closedBall 1 δ = closedBall x δ 
goal: {x} - Metric.closedBall 0 δ = Metric.closedBall x δ

full_name: singleton_div_closedBall_one
code: @[to_additive]
theorem singleton_div_closedBall_one : {x} / closedBall 1 δ = closedBall x δ 
goal: {x} / Metric.closedBall 1 δ = Metric.closedBall x δ

full_name: closedBall_zero_add_singleton
code: @[to_additive]
theorem closedBall_one_mul_singleton : closedBall 1 δ * {x} = closedBall x δ 
goal: Metric.closedBall 0 δ + {x} = Metric.closedBall x δ

full_name: closedBall_one_mul_singleton
code: @[to_additive]
theorem closedBall_one_mul_singleton : closedBall 1 δ * {x} = closedBall x δ 
goal: Metric.closedBall 1 δ * {x} = Metric.closedBall x δ

full_name: closedBall_zero_sub_singleton
code: @[to_additive]
theorem closedBall_one_div_singleton : closedBall 1 δ / {x} = closedBall x⁻¹ δ 
goal: Metric.closedBall 0 δ - {x} = Metric.closedBall (-x) δ

full_name: closedBall_one_div_singleton
code: @[to_additive]
theorem closedBall_one_div_singleton : closedBall 1 δ / {x} = closedBall x⁻¹ δ 
goal: Metric.closedBall 1 δ / {x} = Metric.closedBall x⁻¹ δ

full_name: vadd_closedBall_zero
code: @[to_additive (attr := simp 1100)]
theorem smul_closedBall_one : x • closedBall (1 : E) δ = closedBall x δ 
goal: x +ᵥ Metric.closedBall 0 δ = Metric.closedBall x δ

full_name: smul_closedBall_one
code: @[to_additive (attr := simp 1100)]
theorem smul_closedBall_one : x • closedBall (1 : E) δ = closedBall x δ 
goal: x • Metric.closedBall 1 δ = Metric.closedBall x δ

full_name: mul_ball_one
code: @[to_additive]
theorem mul_ball_one : s * ball 1 δ = thickening δ s 
goal: s * Metric.ball 1 δ = Metric.thickening δ s

full_name: add_ball_zero
code: @[to_additive]
theorem mul_ball_one : s * ball 1 δ = thickening δ s 
goal: s + Metric.ball 0 δ = Metric.thickening δ s

full_name: div_ball_one
code: @[to_additive]
theorem div_ball_one : s / ball 1 δ = thickening δ s 
goal: s / Metric.ball 1 δ = Metric.thickening δ s

full_name: sub_ball_zero
code: @[to_additive]
theorem div_ball_one : s / ball 1 δ = thickening δ s 
goal: s - Metric.ball 0 δ = Metric.thickening δ s

full_name: ball_mul_one
code: @[to_additive]
theorem ball_mul_one : ball 1 δ * s = thickening δ s 
goal: Metric.ball 1 δ * s = Metric.thickening δ s

full_name: ball_add_zero
code: @[to_additive]
theorem ball_mul_one : ball 1 δ * s = thickening δ s 
goal: Metric.ball 0 δ + s = Metric.thickening δ s

full_name: ball_sub_zero
code: @[to_additive]
theorem ball_div_one : ball 1 δ / s = thickening δ s⁻¹ 
goal: Metric.ball 0 δ - s = Metric.thickening δ (-s)

full_name: ball_div_one
code: @[to_additive]
theorem ball_div_one : ball 1 δ / s = thickening δ s⁻¹ 
goal: Metric.ball 1 δ / s = Metric.thickening δ s⁻¹

full_name: mul_ball
code: @[to_additive (attr := simp)]
theorem mul_ball : s * ball x δ = x • thickening δ s 
goal: s * Metric.ball x δ = x • Metric.thickening δ s

full_name: add_ball
code: @[to_additive (attr := simp)]
theorem mul_ball : s * ball x δ = x • thickening δ s 
goal: s + Metric.ball x δ = x +ᵥ Metric.thickening δ s

full_name: div_ball
code: @[to_additive (attr := simp)]
theorem div_ball : s / ball x δ = x⁻¹ • thickening δ s 
goal: s / Metric.ball x δ = x⁻¹ • Metric.thickening δ s

full_name: sub_ball
code: @[to_additive (attr := simp)]
theorem div_ball : s / ball x δ = x⁻¹ • thickening δ s 
goal: s - Metric.ball x δ = -x +ᵥ Metric.thickening δ s

full_name: ball_add
code: @[to_additive (attr := simp)]
theorem ball_mul : ball x δ * s = x • thickening δ s 
goal: Metric.ball x δ + s = x +ᵥ Metric.thickening δ s

full_name: ball_mul
code: @[to_additive (attr := simp)]
theorem ball_mul : ball x δ * s = x • thickening δ s 
goal: Metric.ball x δ * s = x • Metric.thickening δ s

full_name: ball_div
code: @[to_additive (attr := simp)]
theorem ball_div : ball x δ / s = x • thickening δ s⁻¹ 
goal: Metric.ball x δ / s = x • Metric.thickening δ s⁻¹

full_name: ball_sub
code: @[to_additive (attr := simp)]
theorem ball_div : ball x δ / s = x • thickening δ s⁻¹ 
goal: Metric.ball x δ - s = x +ᵥ Metric.thickening δ (-s)

full_name: IsCompact.mul_closedBall_one
code: @[to_additive]
theorem IsCompact.mul_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s * closedBall (1 : E) δ = cthickening δ s 
goal: s * Metric.closedBall 1 δ = Metric.cthickening δ s

full_name: IsCompact.add_closedBall_zero
code: @[to_additive]
theorem IsCompact.mul_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s * closedBall (1 : E) δ = cthickening δ s 
goal: s + Metric.closedBall 0 δ = Metric.cthickening δ s

full_name: IsCompact.div_closedBall_one
code: @[to_additive]
theorem IsCompact.div_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s / closedBall 1 δ = cthickening δ s 
goal: s / Metric.closedBall 1 δ = Metric.cthickening δ s

full_name: IsCompact.sub_closedBall_zero
code: @[to_additive]
theorem IsCompact.div_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s / closedBall 1 δ = cthickening δ s 
goal: s - Metric.closedBall 0 δ = Metric.cthickening δ s

full_name: IsCompact.closedBall_one_mul
code: @[to_additive]
theorem IsCompact.closedBall_one_mul (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ * s = cthickening δ s 
goal: Metric.closedBall 1 δ * s = Metric.cthickening δ s

full_name: IsCompact.closedBall_zero_add
code: @[to_additive]
theorem IsCompact.closedBall_one_mul (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ * s = cthickening δ s 
goal: Metric.closedBall 0 δ + s = Metric.cthickening δ s

full_name: IsCompact.closedBall_one_div
code: @[to_additive]
theorem IsCompact.closedBall_one_div (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ / s = cthickening δ s⁻¹ 
goal: Metric.closedBall 1 δ / s = Metric.cthickening δ s⁻¹

full_name: IsCompact.closedBall_zero_sub
code: @[to_additive]
theorem IsCompact.closedBall_one_div (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ / s = cthickening δ s⁻¹ 
goal: Metric.closedBall 0 δ - s = Metric.cthickening δ (-s)

full_name: IsCompact.mul_closedBall
code: @[to_additive]
theorem IsCompact.mul_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s * closedBall x δ = x • cthickening δ s 
goal: s * Metric.closedBall x δ = x • Metric.cthickening δ s

full_name: IsCompact.add_closedBall
code: @[to_additive]
theorem IsCompact.mul_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s * closedBall x δ = x • cthickening δ s 
goal: s + Metric.closedBall x δ = x +ᵥ Metric.cthickening δ s

full_name: IsCompact.sub_closedBall
code: @[to_additive]
theorem IsCompact.div_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s / closedBall x δ = x⁻¹ • cthickening δ s 
goal: s - Metric.closedBall x δ = -x +ᵥ Metric.cthickening δ s

full_name: IsCompact.div_closedBall
code: @[to_additive]
theorem IsCompact.div_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s / closedBall x δ = x⁻¹ • cthickening δ s 
goal: s / Metric.closedBall x δ = x⁻¹ • Metric.cthickening δ s

full_name: IsCompact.closedBall_add
code: @[to_additive]
theorem IsCompact.closedBall_mul (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
goal: Metric.closedBall x δ + s = x +ᵥ Metric.cthickening δ s

full_name: IsCompact.closedBall_mul
code: @[to_additive]
theorem IsCompact.closedBall_mul (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
goal: Metric.closedBall x δ * s = x • Metric.cthickening δ s

full_name: IsCompact.closedBall_sub
code: @[to_additive]
theorem IsCompact.closedBall_div (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
goal: Metric.closedBall x δ + s = x +ᵥ Metric.cthickening δ s

full_name: IsCompact.closedBall_div
code: @[to_additive]
theorem IsCompact.closedBall_div (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
goal: Metric.closedBall x δ * s = x • Metric.cthickening δ s