File: Mathlib/Analysis/Analytic/Within.lean

full_name: hasFPowerSeriesWithinOnBall_univ
code: @[simp] lemma hasFPowerSeriesWithinOnBall_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {x : E} {r : ℝ≥0∞} :
    HasFPowerSeriesWithinOnBall f p univ x r ↔ HasFPowerSeriesOnBall f p x r 
goal: HasFPowerSeriesWithinOnBall f p Set.univ x r ↔ HasFPowerSeriesOnBall f p x r

full_name: hasFPowerSeriesWithinAt_univ
code: @[simp] lemma hasFPowerSeriesWithinAt_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} :
    HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x 
goal: HasFPowerSeriesWithinAt f p Set.univ x ↔ HasFPowerSeriesAt f p x

full_name: analyticWithinAt_univ
code: @[simp] lemma analyticWithinAt_univ {f : E → F} {x : E} :
    AnalyticWithinAt 𝕜 f univ x ↔ AnalyticAt 𝕜 f x 
goal: AnalyticWithinAt 𝕜 f Set.univ x ↔ AnalyticAt 𝕜 f x

full_name: analyticWithinOn_univ
code: lemma analyticWithinOn_univ {f : E → F} :
    AnalyticWithinOn 𝕜 f univ ↔ AnalyticOn 𝕜 f univ 
goal: AnalyticWithinOn 𝕜 f Set.univ ↔ AnalyticOn 𝕜 f Set.univ

full_name: HasFPowerSeriesWithinAt.continuousWithinAt
code: lemma HasFPowerSeriesWithinAt.continuousWithinAt {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p s x) : ContinuousWithinAt f s x 
goal: ContinuousWithinAt f s x

full_name: AnalyticWithinAt.continuousWithinAt
code: lemma AnalyticWithinAt.continuousWithinAt {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) : ContinuousWithinAt f s x 
goal: ContinuousWithinAt f s x

full_name: analyticWithinAt_of_singleton_mem
code: /-- `AnalyticWithinAt` is trivial if `{x} ∈ 𝓝[s] x` -/
lemma analyticWithinAt_of_singleton_mem {f : E → F} {s : Set E} {x : E} (h : {x} ∈ 𝓝[s] x) :
    AnalyticWithinAt 𝕜 f s x 
goal: AnalyticWithinAt 𝕜 f s x

full_name: AnalyticAt.analyticWithinAt
code: /-- Analyticity implies analyticity within any `s` -/
lemma AnalyticAt.analyticWithinAt {f : E → F} {s : Set E} {x : E} (h : AnalyticAt 𝕜 f x) :
    AnalyticWithinAt 𝕜 f s x 
goal: AnalyticWithinAt 𝕜 f s x

full_name: AnalyticOn.analyticWithinOn
code: /-- Analyticity on `s` implies analyticity within `s` -/
lemma AnalyticOn.analyticWithinOn {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s) :
    AnalyticWithinOn 𝕜 f s 
goal: AnalyticWithinOn 𝕜 f s

full_name: AnalyticWithinOn.continuousOn
code: lemma AnalyticWithinOn.continuousOn {f : E → F} {s : Set E} (h : AnalyticWithinOn 𝕜 f s) :
    ContinuousOn f s 
goal: ContinuousOn f s

full_name: analyticWithinOn_of_locally_analyticWithinOn
code: /-- If `f` is `AnalyticWithinOn` near each point in a set, it is `AnalyticWithinOn` the set -/
lemma analyticWithinOn_of_locally_analyticWithinOn {f : E → F} {s : Set E}
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ AnalyticWithinOn 𝕜 f (s ∩ u)) :
    AnalyticWithinOn 𝕜 f s 
goal: AnalyticWithinOn 𝕜 f s

full_name: IsOpen.analyticWithinOn_iff_analyticOn
code: /-- On open sets, `AnalyticOn` and `AnalyticWithinOn` coincide -/
@[simp] lemma IsOpen.analyticWithinOn_iff_analyticOn {f : E → F} {s : Set E} (hs : IsOpen s) :
    AnalyticWithinOn 𝕜 f s ↔ AnalyticOn 𝕜 f s 
goal: AnalyticWithinOn 𝕜 f s ↔ AnalyticOn 𝕜 f s

full_name: hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall
code: /-- `f` has power series `p` at `x` iff some local extension of `f` has that series -/
lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpace F] {f : E → F}
    {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞} :
    HasFPowerSeriesWithinOnBall f p s x r ↔
      ContinuousWithinAt f s x ∧ ∃ g, EqOn f g (s ∩ EMetric.ball x r) ∧
        HasFPowerSeriesOnBall g p x r 
goal: HasFPowerSeriesWithinOnBall f p s x r ↔
  ContinuousWithinAt f s x ∧ ∃ g, Set.EqOn f g (s ∩ EMetric.ball x r) ∧ HasFPowerSeriesOnBall g p x r

full_name: hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt
code: /-- `f` has power series `p` at `x` iff some local extension of `f` has that series -/
lemma hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt [CompleteSpace F] {f : E → F}
    {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} :
    HasFPowerSeriesWithinAt f p s x ↔
      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ HasFPowerSeriesAt g p x 
goal: HasFPowerSeriesWithinAt f p s x ↔ ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ HasFPowerSeriesAt g p x

full_name: analyticWithinAt_iff_exists_analyticAt
code: /-- `f` is analytic within `s` at `x` iff some local extension of `f` is analytic at `x` -/
lemma analyticWithinAt_iff_exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E} :
    AnalyticWithinAt 𝕜 f s x ↔
      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x 
goal: AnalyticWithinAt 𝕜 f s x ↔ ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x

full_name: AnalyticWithinAt.exists_analyticAt
code: /-- If `f` is analytic within `s` at `x`, some local extension of `f` is analytic at `x` -/
lemma AnalyticWithinAt.exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) : ∃ g, f x = g x ∧ f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x 
goal: ∃ g, f x = g x ∧ f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x

full_name: AnalyticWithinAt.congr_of_eventuallyEq
code: lemma AnalyticWithinAt.congr_of_eventuallyEq [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
    (hf : AnalyticWithinAt 𝕜 f s x) (hs : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
    AnalyticWithinAt 𝕜 g s x 
goal: AnalyticWithinAt 𝕜 g s x

full_name: AnalyticWithinAt.congr
code: lemma AnalyticWithinAt.congr [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
    (hf : AnalyticWithinAt 𝕜 f s x) (hs : EqOn f g s) (hx : f x = g x) :
    AnalyticWithinAt 𝕜 g s x 
goal: AnalyticWithinAt 𝕜 g s x

full_name: AnalyticWithinOn.congr
code: lemma AnalyticWithinOn.congr [CompleteSpace F] {f g : E → F} {s : Set E}
    (hf : AnalyticWithinOn 𝕜 f s) (hs : EqOn f g s) :
    AnalyticWithinOn 𝕜 g s 
goal: AnalyticWithinOn 𝕜 g s

full_name: HasFPowerSeriesWithinOnBall.mono
code: lemma HasFPowerSeriesWithinOnBall.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s t : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p t x r)
    (hs : s ⊆ t) : HasFPowerSeriesWithinOnBall f p s x r 
goal: HasFPowerSeriesWithinOnBall f p s x r

full_name: HasFPowerSeriesWithinAt.mono
code: lemma HasFPowerSeriesWithinAt.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s t : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p t x)
    (hs : s ⊆ t) : HasFPowerSeriesWithinAt f p s x 
goal: HasFPowerSeriesWithinAt f p s x

full_name: AnalyticWithinAt.mono
code: lemma AnalyticWithinAt.mono {f : E → F} {s t : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f t x)
    (hs : s ⊆ t) : AnalyticWithinAt 𝕜 f s x 
goal: AnalyticWithinAt 𝕜 f s x

full_name: AnalyticWithinOn.mono
code: lemma AnalyticWithinOn.mono {f : E → F} {s t : Set E} (h : AnalyticWithinOn 𝕜 f t)
    (hs : s ⊆ t) : AnalyticWithinOn 𝕜 f s 
goal: AnalyticWithinOn 𝕜 f s

full_name: AnalyticWithinAt.comp
code: lemma AnalyticWithinAt.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
    {t : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s (g x)) (hg : AnalyticWithinAt 𝕜 g t x)
    (h : MapsTo g t s) : AnalyticWithinAt 𝕜 (f ∘ g) t x 
goal: AnalyticWithinAt 𝕜 (f ∘ g) t x

full_name: AnalyticWithinOn.comp
code: lemma AnalyticWithinOn.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
    {t : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g t) (h : MapsTo g t s) :
    AnalyticWithinOn 𝕜 (f ∘ g) t 
goal: AnalyticWithinOn 𝕜 (f ∘ g) t

full_name: AnalyticWithinAt.contDiffWithinAt
code: lemma AnalyticWithinAt.contDiffWithinAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) {n : ℕ∞} : ContDiffWithinAt 𝕜 n f s x 
goal: ContDiffWithinAt 𝕜 n f s x

full_name: AnalyticWithinOn.contDiffOn
code: lemma AnalyticWithinOn.contDiffOn [CompleteSpace F] {f : E → F} {s : Set E}
    (h : AnalyticWithinOn 𝕜 f s) {n : ℕ∞} : ContDiffOn 𝕜 n f s 
goal: ContDiffOn 𝕜 n f s

full_name: HasFPowerSeriesWithinOnBall.prod
code: lemma HasFPowerSeriesWithinOnBall.prod {e : E} {f : E → F} {g : E → G} {s : Set E} {r t : ℝ≥0∞}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesWithinOnBall f p s e r) (hg : HasFPowerSeriesWithinOnBall g q s e t) :
    HasFPowerSeriesWithinOnBall (fun x ↦ (f x, g x)) (p.prod q) s e (min r t) 
goal: HasFPowerSeriesWithinOnBall (fun x => (f x, g x)) (p.prod q) s e (min r t)

full_name: HasFPowerSeriesWithinAt.prod
code: lemma HasFPowerSeriesWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesWithinAt f p s e) (hg : HasFPowerSeriesWithinAt g q s e) :
    HasFPowerSeriesWithinAt (fun x ↦ (f x, g x)) (p.prod q) s e 
goal: HasFPowerSeriesWithinAt (fun x => (f x, g x)) (p.prod q) s e

full_name: AnalyticWithinAt.prod
code: lemma AnalyticWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
    (hf : AnalyticWithinAt 𝕜 f s e) (hg : AnalyticWithinAt 𝕜 g s e) :
    AnalyticWithinAt 𝕜 (fun x ↦ (f x, g x)) s e 
goal: AnalyticWithinAt 𝕜 (fun x => (f x, g x)) s e

full_name: AnalyticWithinOn.prod
code: lemma AnalyticWithinOn.prod {f : E → F} {g : E → G} {s : Set E}
    (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g s) :
    AnalyticWithinOn 𝕜 (fun x ↦ (f x, g x)) s 
goal: AnalyticWithinOn 𝕜 (fun x => (f x, g x)) s