Theorem: Complex.abs_circleTransformBoundingFunction_le
File: Mathlib/MeasureTheory/Integral/CircleTransform.lean

Proof steps:
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
tactic: have cts := continuousOn_abs_circleTransformBoundingFunction hr z
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
tactic: have comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]]) := by
    apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
tactic: have none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty :=
    (nonempty_closedBall.2 hr').prod nonempty_uIcc
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty
this :
  ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],
    IsMaxOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ [[0, 2 * π]]) x
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
tactic: have := IsCompact.exists_isMaxOn comp none (cts.mono <| prod_mono_right (subset_univ _))
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
cts : ContinuousOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])
none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty
this :
  ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],
    IsMaxOn (⇑abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ [[0, 2 * π]]) x
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      abs (circleTransformBoundingFunction R z ↑y) ≤ abs (circleTransformBoundingFunction R z ↑x)
state_after: no goals
tactic: simpa [isMaxOn_iff] using this