state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case h₀
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
e : E
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical | Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s✝ : Finset ι
e : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.sup'_cons hs, Finset.inf'_cons hs, ball_sup] | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical simp
· | Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s✝ : Finset ι
e : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [inf_eq_inter, ih] | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical simp
· rw [Finset.sup'_cons hs, Finset.inf'_cons hs, ball_sup]
... | Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
H : Finset.Non... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
| Mathlib.Analysis.Seminorm.768_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case h₀
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
e : E
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | classical simp | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· | Mathlib.Analysis.Seminorm.768_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case h₀
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
e : E
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical | Mathlib.Analysis.Seminorm.768_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s✝ : Finset ι
e : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.sup'_cons hs, Finset.inf'_cons hs, closedBall_sup] | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical simp
· | Mathlib.Analysis.Seminorm.768_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s✝ : Finset ι
e : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [inf_eq_inter, ih] | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· classical simp
· rw [Finset.sup'_cons hs, Finset.inf'_cons... | Mathlib.Analysis.Seminorm.768_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E)
(r : ℝ) : closedBall (s.sup' H p) e r = s.inf' H fun i => closedBall (p i) e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ r₂ : ℝ
x₁ x₂ : E
⊢ ball p x₁ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩ | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) := by
| Mathlib.Analysis.Seminorm.793_0.ywwMCgoKeIFKDZ3 | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
case intro.intro.intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_ball, add_sub_add_comm] | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩
| Mathlib.Analysis.Seminorm.793_0.ywwMCgoKeIFKDZ3 | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
case intro.intro.intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact (map_add_le_add p _ _).trans_lt (add_lt_add hy₁ hy₂) | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩
rw [mem_ball, add_sub_add_comm]
| Mathlib.Analysis.Seminorm.793_0.ywwMCgoKeIFKDZ3 | theorem ball_add_ball_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ r₂ : ℝ
x₁ x₂ : E
⊢ closedBall... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩ | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) := by
| Mathlib.Analysis.Seminorm.800_0.ywwMCgoKeIFKDZ3 | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
case intro.intro.intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_closedBall, add_sub_add_comm] | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩
| Mathlib.Analysis.Seminorm.800_0.ywwMCgoKeIFKDZ3 | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
case intro.intro.intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r : ℝ
p : Seminorm 𝕜 E
r₁ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact (map_add_le_add p _ _).trans (add_le_add hy₁ hy₂) | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, y₂, hy₁, hy₂, rfl⟩
rw [mem_closedBall, add_sub_add_comm]
| Mathlib.Analysis.Seminorm.800_0.ywwMCgoKeIFKDZ3 | theorem closedBall_add_closedBall_subset (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E) :
p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y✝ : E
r✝ : ℝ
p : Seminorm 𝕜 E
x₁ x₂ y : E
r : ℝ
⊢ x₁ - x₂ ∈ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [mem_ball, sub_sub] | theorem sub_mem_ball (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r := by | Mathlib.Analysis.Seminorm.807_0.ywwMCgoKeIFKDZ3 | theorem sub_mem_ball (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r := by
| Mathlib.Analysis.Seminorm.833_0.ywwMCgoKeIFKDZ3 | theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [ball, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub] | theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r := by
ext
| Mathlib.Analysis.Seminorm.833_0.ywwMCgoKeIFKDZ3 | theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r := by
| Mathlib.Analysis.Seminorm.839_0.ywwMCgoKeIFKDZ3 | theorem closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [closedBall, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub] | theorem closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r := by
ext
| Mathlib.Analysis.Seminorm.839_0.ywwMCgoKeIFKDZ3 | theorem closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext x | theorem preimage_metric_ball {r : ℝ} : p ⁻¹' Metric.ball 0 r = { x | p x < r } := by
| Mathlib.Analysis.Seminorm.847_0.ywwMCgoKeIFKDZ3 | theorem preimage_metric_ball {r : ℝ} : p ⁻¹' Metric.ball 0 r = { x | p x < r } | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [mem_setOf, mem_preimage, mem_ball_zero_iff, Real.norm_of_nonneg (map_nonneg p _)] | theorem preimage_metric_ball {r : ℝ} : p ⁻¹' Metric.ball 0 r = { x | p x < r } := by
ext x
| Mathlib.Analysis.Seminorm.847_0.ywwMCgoKeIFKDZ3 | theorem preimage_metric_ball {r : ℝ} : p ⁻¹' Metric.ball 0 r = { x | p x < r } | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext x | theorem preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } := by
| Mathlib.Analysis.Seminorm.852_0.ywwMCgoKeIFKDZ3 | theorem preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [mem_setOf, mem_preimage, mem_closedBall_zero_iff,
Real.norm_of_nonneg (map_nonneg p _)] | theorem preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } := by
ext x
| Mathlib.Analysis.Seminorm.852_0.ywwMCgoKeIFKDZ3 | theorem preimage_metric_closedBall {r : ℝ} : p ⁻¹' Metric.closedBall 0 r = { x | p x ≤ r } | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [ball_zero_eq, preimage_metric_ball] | theorem ball_zero_eq_preimage_ball {r : ℝ} : p.ball 0 r = p ⁻¹' Metric.ball 0 r := by
| Mathlib.Analysis.Seminorm.858_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_eq_preimage_ball {r : ℝ} : p.ball 0 r = p ⁻¹' Metric.ball 0 r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [closedBall_zero_eq, preimage_metric_closedBall] | theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r := by
| Mathlib.Analysis.Seminorm.862_0.ywwMCgoKeIFKDZ3 | theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro a ha x ⟨y, hy, hx⟩ | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by
| Mathlib.Analysis.Seminorm.878_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : M... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_ball_zero, ← hx, map_smul_eq_mul] | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
| Mathlib.Analysis.Seminorm.878_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : M... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | calc
_ ≤ p y := mul_le_of_le_one_left (map_nonneg p _) ha
_ < r := by rwa [mem_ball_zero] at hy | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_ball_zero, ← hx, map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.878_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rwa [mem_ball_zero] at hy | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_ball_zero, ← hx, map_smul_eq_mul]
calc
_ ≤ p y := mul_le_of_le_one_left (map_nonneg p _) ha
_ < r := by | Mathlib.Analysis.Seminorm.878_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls at the origin are balanced. -/
theorem balanced_ball_zero (r : ℝ) : Balanced 𝕜 (ball p 0 r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro a ha x ⟨y, hy, hx⟩ | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) := by
| Mathlib.Analysis.Seminorm.887_0.ywwMCgoKeIFKDZ3 | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : M... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_closedBall_zero, ← hx, map_smul_eq_mul] | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
| Mathlib.Analysis.Seminorm.887_0.ywwMCgoKeIFKDZ3 | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : M... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | calc
_ ≤ p y := mul_le_of_le_one_left (map_nonneg p _) ha
_ ≤ r := by rwa [mem_closedBall_zero] at hy | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_closedBall_zero, ← hx, map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.887_0.ywwMCgoKeIFKDZ3 | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rwa [mem_closedBall_zero] at hy | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) := by
rintro a ha x ⟨y, hy, hx⟩
rw [mem_closedBall_zero, ← hx, map_smul_eq_mul]
calc
_ ≤ p y := mul_le_of_le_one_left (map_nonneg p _) ha
_ ≤ r := by | Mathlib.Analysis.Seminorm.887_0.ywwMCgoKeIFKDZ3 | /-- Closed seminorm-balls at the origin are balanced. -/
theorem balanced_closedBall_zero (r : ℝ) : Balanced 𝕜 (closedBall p 0 r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | lift r to NNReal using hr.le | theorem ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r := by
| Mathlib.Analysis.Seminorm.896_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [ball, iInter_setOf, finset_sup_apply, NNReal.coe_lt_coe,
Finset.sup_lt_iff (show ⊥ < r from hr), ← NNReal.coe_lt_coe, NNReal.coe_mk] | theorem ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r := by
lift r to NNReal using hr.le
| Mathlib.Analysis.Seminorm.896_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 < r) : ball (s.sup p) x r = ⋂ i ∈ s, ball (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | lift r to NNReal using hr | theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r := by
| Mathlib.Analysis.Seminorm.903_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [closedBall, iInter_setOf, finset_sup_apply, NNReal.coe_le_coe, Finset.sup_le_iff, ←
NNReal.coe_le_coe, NNReal.coe_mk] | theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r := by
lift r to NNReal using hr
| Mathlib.Analysis.Seminorm.903_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.inf_eq_iInf] | theorem ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf fun i => ball (p i) x r := by
| Mathlib.Analysis.Seminorm.910_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf fun i => ball (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact ball_finset_sup_eq_iInter _ _ _ hr | theorem ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf fun i => ball (p i) x r := by
rw [Finset.inf_eq_iInf]
| Mathlib.Analysis.Seminorm.910_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf fun i => ball (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.inf_eq_iInf] | theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r := by
| Mathlib.Analysis.Seminorm.916_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact closedBall_finset_sup_eq_iInter _ _ _ hr | theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r := by
rw [Finset.inf_eq_iInf]
| Mathlib.Analysis.Seminorm.916_0.ywwMCgoKeIFKDZ3 | theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ} (hr : 0 ≤ r) :
closedBall (s.sup p) x r = s.inf fun i => closedBall (p i) x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ := by
| Mathlib.Analysis.Seminorm.922_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.mem_ball, Set.mem_empty_iff_false, iff_false_iff, not_lt] | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ := by
ext
| Mathlib.Analysis.Seminorm.922_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact hr.trans (map_nonneg p _) | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ := by
ext
rw [Seminorm.mem_ball, Set.mem_empty_iff_false, iff_false_iff, not_lt]
| Mathlib.Analysis.Seminorm.922_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ := by
| Mathlib.Analysis.Seminorm.929_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false_iff, not_le] | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ := by
ext
| Mathlib.Analysis.Seminorm.929_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact hr.trans_le (map_nonneg _ _) | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ := by
ext
rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false_iff, not_le]
| Mathlib.Analysis.Seminorm.929_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero_iff, map_smul_eq_mul] | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
| Mathlib.Analysis.Seminorm.937_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine fun a ha b hb ↦ mul_lt_mul' ha hb (map_nonneg _ _) ?_ | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.937_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact hr₁.lt_or_lt.resolve_left <| ((norm_nonneg a).trans ha).not_lt | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
refine fun a ha b hb ↦ mul_lt_mul' ha hb (map_nonneg _ _) ?_
| Mathlib.Analysis.Seminorm.937_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul] | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
| Mathlib.Analysis.Seminorm.943_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro a ha b hb | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.943_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mul_comm, mul_comm r₁] | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul]
intro a ha b hb
| Mathlib.Analysis.Seminorm.943_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine mul_lt_mul' hb ha (norm_nonneg _) (hr₂.lt_or_lt.resolve_left ?_) | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul]
intro a ha b hb
rw [mul_comm, mul_comm r₁]
| Mathlib.Analysis.Seminorm.943_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact ((map_nonneg p b).trans hb).not_lt | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_ball_zero, mem_closedBall_zero, mem_ball_zero_iff,
map_smul_eq_mul]
intro a ha b hb
rw [mul_comm, mul_comm r₁]
refine... | Mathlib.Analysis.Seminorm.943_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_closedBall (p : Seminorm 𝕜 E) (r₁ : ℝ) {r₂ : ℝ} (hr₂ : r₂ ≠ 0) :
Metric.ball (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases eq_or_ne r₂ 0 with rfl | hr₂ | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
| Mathlib.Analysis.Seminorm.952_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
rcases eq_or_ne r₂ 0 with rfl | hr₂
· | Mathlib.Analysis.Seminorm.952_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact (smul_subset_smul_left (ball_subset_closedBall _ _ _)).trans
(ball_smul_closedBall _ _ hr₂) | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) := by
rcases eq_or_ne r₂ 0 with rfl | hr₂
· simp
· | Mathlib.Analysis.Seminorm.952_0.ywwMCgoKeIFKDZ3 | theorem ball_smul_ball (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul] | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) := by
| Mathlib.Analysis.Seminorm.960_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro a ha b hb | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.960_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | gcongr | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
intro a ha b hb
| Mathlib.Analysis.Seminorm.960_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
case b0
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact (norm_nonneg _).trans ha | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) := by
simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul]
intro a ha b hb
gcongr
| Mathlib.Analysis.Seminorm.960_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul_closedBall (p : Seminorm 𝕜 E) (r₁ r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.closedBall 0 r₂ ⊆ p.closedBall 0 (r₁ * r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simpa only [mem_ball_zero, map_neg_eq_map] using hx | theorem neg_mem_ball_zero (r : ℝ) (hx : x ∈ ball p 0 r) : -x ∈ ball p 0 r := by
| Mathlib.Analysis.Seminorm.969_0.ywwMCgoKeIFKDZ3 | theorem neg_mem_ball_zero (r : ℝ) (hx : x ∈ ball p 0 r) : -x ∈ ball p 0 r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ E₂
σ₁₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | @[simp]
theorem neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r := by
| Mathlib.Analysis.Seminorm.973_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedRing 𝕜₂
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Set.mem_neg, mem_ball, mem_ball, ← neg_add', sub_neg_eq_add, map_neg_eq_map] | @[simp]
theorem neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r := by
ext
| Mathlib.Analysis.Seminorm.973_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem neg_ball (p : Seminorm 𝕜 E) (r : ℝ) (x : E) : -ball p x r = ball p (-x) r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : ι → Seminorm 𝕜 E
hp : BddAbo... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | cases isEmpty_or_nonempty ι | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
| Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : ι → Seminorm 𝕜 E
hp... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty] | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
cases isEmpty_or_nonempty ι
· | Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : ι → Seminorm 𝕜 E
hp... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact closedBall_bot _ hr | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty]
| Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : ι → Seminorm 𝕜 E
hp... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext x | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty]
exact closedBall_bot _ hr
· | Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case inr.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : ι → Seminorm 𝕜 E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | have := Seminorm.bddAbove_range_iff.mp hp (x - e) | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty]
exact closedBall_bot _ hr
· ext x
| Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
case inr.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : ι → Seminorm 𝕜 E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [mem_closedBall, mem_iInter, Seminorm.iSup_apply hp, ciSup_le_iff this] | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty]
exact closedBall_bot _ hr
· ext x
have := Seminor... | Mathlib.Analysis.Seminorm.990_0.ywwMCgoKeIFKDZ3 | theorem closedBall_iSup {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E) {r : ℝ}
(hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
⊢ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases eq_or_ne k 0 with (rfl | hk) | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
| Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
r : ℝ
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl] | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· | Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
r : ℝ
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact empty_subset _ | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
| Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro x | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· | Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E
k : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Set.mem_smul_set, Seminorm.mem_ball_zero] | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· intro x
| Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E
k : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' fun hx => ⟨k⁻¹ • x, _, _⟩ | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· intro x
rw [Set.mem_smul_set, Seminorm.mem_ball_zero]
| Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inr.refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rwa [Seminorm.mem_ball_zero, map_smul_eq_mul, norm_inv, ←
mul_lt_mul_left <| norm_pos_iff.mpr hk, ← mul_assoc, ← div_eq_mul_inv ‖k‖ ‖k‖,
div_self (ne_of_gt <| norm_pos_iff.mpr hk), one_mul] | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· intro x
rw [Set.mem_smul_set, Seminorm.mem_ball_zero]
refine' fun hx => ⟨k⁻... | Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
case inr.refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← smul_assoc, smul_eq_mul, ← div_eq_mul_inv, div_self hk, one_smul] | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· intro x
rw [Set.mem_smul_set, Seminorm.mem_ball_zero]
refine' fun hx => ⟨k⁻... | Mathlib.Analysis.Seminorm.999_0.ywwMCgoKeIFKDZ3 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
k • p.ball 0 r = p.ball 0 (‖k‖ * r) := by
| Mathlib.Analysis.Seminorm.1013_0.ywwMCgoKeIFKDZ3 | theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
k • p.ball 0 r = p.ball 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_smul_set_iff_inv_smul_mem₀ hk, p.mem_ball_zero, p.mem_ball_zero, map_smul_eq_mul,
norm_inv, ← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hk), mul_comm] | theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
k • p.ball 0 r = p.ball 0 (‖k‖ * r) := by
ext
| Mathlib.Analysis.Seminorm.1013_0.ywwMCgoKeIFKDZ3 | theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
k • p.ball 0 r = p.ball 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
⊢ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro x ⟨y, hy, h⟩ | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) := by
| Mathlib.Analysis.Seminorm.1020_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.mem_closedBall_zero, ← h, map_smul_eq_mul] | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) := by
rintro x ⟨y, hy, h⟩
| Mathlib.Analysis.Seminorm.1020_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.mem_closedBall_zero] at hy | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) := by
rintro x ⟨y, hy, h⟩
rw [Seminorm.mem_closedBall_zero, ← h, map_smul_eq_mul]
| Mathlib.Analysis.Seminorm.1020_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | gcongr | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) := by
rintro x ⟨y, hy, h⟩
rw [Seminorm.mem_closedBall_zero, ← h, map_smul_eq_mul]
rw [Seminorm.mem_closedBall_zero] at hy
| Mathlib.Analysis.Seminorm.1020_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' subset_antisymm smul_closedBall_subset _ | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
| Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro x | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
refine' subset_antisymm smul_closedBall_subset _
| Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
h... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Set.mem_smul_set, Seminorm.mem_closedBall_zero] | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
refine' subset_antisymm smul_closedBall_subset _
intro x
| Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
h... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' fun hx => ⟨k⁻¹ • x, _, _⟩ | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
refine' subset_antisymm smul_closedBall_subset _
intro x
rw [Set.mem_smul_set, Seminorm.mem_closedBall_zero]
| Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rwa [Seminorm.mem_closedBall_zero, map_smul_eq_mul, norm_inv, ← mul_le_mul_left hk, ← mul_assoc,
← div_eq_mul_inv ‖k‖ ‖k‖, div_self (ne_of_gt hk), one_mul] | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
refine' subset_antisymm smul_closedBall_subset _
intro x
rw [Set.mem_smul_set, Seminorm.mem_closedBall_zero]
refine' fun hx => ⟨k⁻¹ • x, _, _⟩
· | Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← smul_assoc, smul_eq_mul, ← div_eq_mul_inv, div_self (norm_pos_iff.mp hk), one_smul] | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) := by
refine' subset_antisymm smul_closedBall_subset _
intro x
rw [Set.mem_smul_set, Seminorm.mem_closedBall_zero]
refine' fun hx => ⟨k⁻¹ • x, _, _⟩
· rwa [Seminorm.mem_closed... | Mathlib.Analysis.Seminorm.1028_0.ywwMCgoKeIFKDZ3 | theorem smul_closedBall_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closedBall 0 r = p.closedBall 0 (‖k‖ * r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r : ℝ
x : E
p : Seminorm 𝕜 E
r₁ r₂ : ℝ
hr₁ : ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩ | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
| Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' ⟨r, hr₀, fun a ha x hx => _⟩ | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
| Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a✝ : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero] | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
refine' ⟨r, hr₀, fun a ha x hx => _⟩
| Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a✝ : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [p.mem_ball_zero] at hx | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
refine' ⟨r, hr₀, fun a ha x hx => _⟩
rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]
| Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a✝ : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact hx.trans (hr.trans_le <| by gcongr) | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
refine' ⟨r, hr₀, fun a ha x hx => _⟩
rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]
rw [p.mem_ball_zer... | Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a✝ : 𝕜
r✝ : ℝ
x✝ : E
p : Seminorm 𝕜 E
r₁ r₂ : ℝ
hr₁... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | gcongr | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
refine' ⟨r, hr₀, fun a ha x hx => _⟩
rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]
rw [p.mem_ball_zer... | Mathlib.Analysis.Seminorm.1039_0.ywwMCgoKeIFKDZ3 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r : ℝ
x : E
hpr : p x < r
⊢ Absorbent 𝕜 (ball ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' (p.absorbent_ball_zero <| sub_pos.2 hpr).subset fun y hy => _ | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) := by
| Mathlib.Analysis.Seminorm.1060_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r : ℝ
x : E
hpr : p x < r
y : E
hy : y ∈ ball p... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [p.mem_ball_zero] at hy | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) := by
refine' (p.absorbent_ball_zero <| sub_pos.2 hpr).subset fun y hy => _
| Mathlib.Analysis.Seminorm.1060_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r : ℝ
x : E
hpr : p x < r
y : E
hy : p y < r - ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact p.mem_ball.2 ((map_sub_le_add p _ _).trans_lt <| add_lt_of_lt_sub_right hy) | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) := by
refine' (p.absorbent_ball_zero <| sub_pos.2 hpr).subset fun y hy => _
rw [p.mem_ball_zero] at hy
| Mathlib.Analysis.Seminorm.1060_0.ywwMCgoKeIFKDZ3 | /-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) | Mathlib_Analysis_Seminorm |
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